THE 


•SENIOR  ARITHMETIC 


» 
FOlt 


GRAMMAR    SCHOOLS 


BY 

CHARLES  E.  WHITE 

PRINCIPAL  FKANKLIX  SCHOOL,  SYRA(  i  ,SK. 


BOSTON,  U.S.A. 

IX    C.    HEATH    &    CO.,    PUBLISHERS 
1897 


COPYRIGHT,  1897, 
BY  D.  C.  HEATH  &  CO. 


TYPOGRAPHY  UY  C.  J.  PETERS  &  SON,  BOSTON. 


PBLSSWOKK  BY  ROCKWELL  &  CHUKCHI LL. 


PREFACE. 


THE  SEXIOK  ARITHMETIC  is  intended  for  use  in  the 
higher  grammar  grades,  beginning  with  the  sixth  year. 

The  first  eight  pages  are  devoted  to  definitions  in  review, 
covering  fundamental  subjects  which  should  be  mastered 
before  the  book  is  taken  up.  The  senior  arithmetic  proper 
begins  with  decimals.  Though  largely  review  matter, 
denominate  numbers  are  treated  with  considerable  fulness. 
Teachers  are  expected,  however,  to  use  their  discretion 
as  to  omissions  in  this  subject. 

It  has  been  the  author's  aim  to  employ  such  definitions, 
solutions,  explanations,  and  rules  as  can  be  readily  com- 
prehended and  applied  by  the  pupils,  unaided  by  the 
teacher.  While  care  has  been  exercised  in  selecting  a 
great  variety  of  practical  business  problems  and  in  arran- 
ging them  progressively,  the  development  of  mental  power 
has  been  kept  constantly  in  view. 

The  arrangement  of  this  book  is  topical,  but  subjects 
previously  studied  are  kept  fresh  in  the  minds  of  the 
pupils  by  frequent  carefully  prepared  reviews. 

The  practice  of  referring  percentage  problems  back  to 
the  original  questions  of  relation  has  proven  highly  suc- 
cessful in  the  author's  experience. 

85984 


IV  VKKFAOK. 

Thanks  are  due  to  the  various  superintendents  of  city 
schools  who  kindly  furnished  copies  of  recent  examination 
questions,  which  largely  constitute  the  test  problems  of 
this  book. 

The  author  has  also  received  invaluable  aid  from  many 
leading  educators  in  the  State  of  New  York,  all  of  whom 
he  desires  to  thank  most  cordially. 

C.  E.  W. 

SYRACUSE,  N.Y.,  Oct.  18,  18%. 


CONTENTS. 


DEFINITIONS,  RULES,  AND  PRINCIPLES  IN  REVIEW —  PAGE 

Notation,  Numeration 1 

Addition,  Subtraction,  Multiplication 2 

Division,  Factors,  Multiples     .* 3 

Least  Common  Multiple 4 

Greatest  Common  Divisor 4 

Cancellation 5 

Common  Fractions 5 

Addition  arid  Subtraction  of  Fractions 7 

Multiplication  and  Division  of  Fractions 8 

DECIMAL  FRACTIONS  — 

To  read  a  decimal 9 

To  write  a  decimal 10 

To  reduce  two  or  more  decimals  to  a  common  denominator  .     .  11 

To  reduce  a  common  fraction  to  a  decimal 12 

Addition  of  decimals 13 

Subtraction  of  decimals 13 

Multiplication  of  decimals 14 

To  multiply  by  10,  100,  1000,  etc 15 

Division  of  decimals 16 

To  divide  by  10,  100,  1000,  etc 1.7 

Parts  of  100  and  1000 18 

To  multiply  by  25 18 

Aliquot  Parts  of  a  dollar 19 

Review  of  decimals 21 

Accounts  and  Bills 25 

Indicated  Operations "27 

MISCELLANEOUS  REVIEW  — 

Factors,  Multiples,  Divisors,  and  Cancellation 30 

Common  Fractions 32 

Review  questions 39 

COMPOUND  NUMBERS  — 

Linear  measure 43 

Surveyors'  measure 44 

v 


VI  CONTENTS. 

COMPOUND  NUMBERS  (Continued) —  PAGE 

Square  measure 44 

Cubic  measure 44 

Liquid  measure 45 

Apothecaries'  fluid  measure 45 

Dry  measure 45 

Avoirdupois  weight 45 

Troy  weight 46 

Apothecaries'  weight 46 

Measure  of  time 46 

Circular  measure 47 

Federal  money 48 

English  or  Sterling  money 48 

Counting  table 49 

Paper  table 49 

Reduction  descending 49 

Reduction  ascending 51 

Review  problems 53 

To  reduce  denominate  fractions  to  integers  of  lower  denomi- 
nations    55 

To  reduce  denominate  numbers  to  fractions  of  higher  denomi- 
nations    56 

To  find  what  part  one  denominate  number  is  of  another    .     .     .  .">« 

Addition  of  compound  numbers 58 

Subtraction  of  compound  numbers 61 

Difference  between  dates 63 

Multiplication  of  compound  numbers 64 

Division  of  compound  numbers 65 

Miscellaneous  problems 69 

MEASUREMENTS,  Surfaces 72 

Carpeting  rooms 77 

Plastering  and  painting 79 

Papering  walls 81 

Board  measure 82 

Miscellaneous  problems 84 

MEASUREMENTS,  Solids 86 

Wood  measure 89 

Capacity  of  bins 90 

LONGITUDE  AND  TIME 91 

Standard  time 92 

Review  questions 94 

THE  METRIC  SYSTEM  — 

Linear  measure 96 

Surface  measure 100 


CONTENTS.  Vll 

THE  METRIC  SYSTEM  (Continued) —  PAGE 

Volume  measure 102 

Capacity  measure 104 

Measure  of  weight   .     .     .    ' 105 

Review  questions 10j5 

General  review 107 

PERCENTAGE 114 

Profit  and  loss 124 

Commission 12(J 

Insurance 130 

Trade  discount 132 

Taxes 134 

Duties 13(5 

Review  questions 137 

Miscellaneous  review  of  percentage 137 

SIMPLE  INTEREST 143 

Exact  interest 149 

To  find  the  rate,  when  principal,  interest,  and  time  are  given    .  150 

To  find  time,  when  principal,  interest,  and  rate  are  given      .     .  151 
To  find  principal,  when  interest  or  amount,  rate  and  time  are 

given 152 

Promissory  notes 153 

Partial  payments,  U.  S.  rule 156 

Merchants'  rule 159 

Compound  interest 160 

Review  of  interest 161 

TRUE  DISCOUNT 165 

BANK  DISCOUNT 167 

To  find  the  face  of  a  note  when  the  proceeds,  time,  and  rate  are 

known 173 

Review  of  discount 174 

STOCKS  AND  BONDS 176 

Bonds 179 

Miscellaneous .....' 182 

AVERAGE  OF  PAYMENTS 183 

Review  questions 188 

RATIO  AND  PROPORTION  — 

Ratio 189 

Simple  proportion 190 

Compound  proportion 195 

Partnership 198 

Review  questions 202 


Vlll  CONTKNTS. 

INVOLUTION  AND  EVOLUTION —  PACK 

Involution 203 

Evolution 204 

Square  root • 205 

.     Right-angled  triangles 211 

Similar  surfaces 213 

Cube  root 214 

Similar  solids 220 

Questions 221 

General  review 222 

TEST  QUESTIONS 236 

Decimals 244 

Denominate  numbers 248 

Percentage 255 

Interest  and  discount 263 

Proportion  and  partnership 271 

Involution  and  evolution 274 

Miscellaneous 276 

MENSURATION  — 

Surfaces 285 

Solids 287 

Review  of  mensuration    .                                                                   .  290 


SENIOR   ARITHMETIC. 


DEFINITIONS. 

1.  A  Unit  is  one,  or  one  thing. 

2.  A  Number  is  that  which  tells  how  many. 

3.  The  Unit  of  a  Number  is  one  of  its  units. 

4.  Numbers  having  the  same  unit  are  Like  Numbers. 

5.  A  number  not  applied   to  any  particular  object  is 
an  Abstract  Number ;  as  6,  11,  15. 

6.  A  number  that  is  applied  to  a  particular  object  is 
a  Concrete  Number;  as  6  men,  11  IV)..  1T>  days. 

7.  An  Integer  is  a  whole  number. 

8.  Expressing  numbers  by  figures  or  letters  is  called 
Notation. 

9.  Arabic  Notation  is  expressing  numbers  by  figures. 

10.  Roman  Notation  is  expressing  numbers  by  letters. 

11.  Naming  the  places  of  figures  and  reading  numbers 
is  Numeration. 

12.  A  figure  standing  alone  expresses  units. 

13.  When  figures   stand   side   by   side,    the   right-hand 
figure  expresses   units,  the  next  tens,  the  next  hundreds, 
etc. 

1 


2  SENIOR    A  KITH  MET  1C. 

14.  The  value  of  a  figure,  without  regard  to  its  place,  is 
its  Simple  Value.     The  value  of  a  figure  with  reference  to 
its  place  in  a  number  is  its  Local  Value. 

NOTE.  —  In  the  number  5555,  the  simple  value  of  each  figure  is  5. 
The  local  value  of  the  right-hand  figure  is  5.  Of  the  second,  50.  Of 
the  third,  500.  Of  the  fourth,  5000. 

15.  Each  group  of  three  figures,  beginning  with  units 
and  counting  to  the  left,  is  a  Period. 

TO   READ   NUMBERS. 

Rule.  — Begin  at  tJie  right,  and  separate  the  numbers  into 
groups  of  three  figures  each,  using  the  comma. 

Begin  at  the  left,  and  read  the  number  in  each  group,  giv- 
ing to  it  the  name  of  that  group, 

Xo  name  is  given  to  the  number  in  the  last  f/ro///>. 

16.  Addition  is  the  process  of  uniting  two  or  more  like 
numbers  into  one  sum. 

17.  The  result  of  addition  is  called  the  Sum  or  Amount. 

18.  Subtraction  is  the  process  of  finding  the  difference 
between  two  like  numbers. 

19.  The  number  from  which  we  subtract  is  called  the 
Minuend,  and  the  number  subtracted,  the  Subtrahend.     The 
result  in  subtraction  is  the  Difference  or  Remainder. 

20.  Multiplication   is   the   process   of    finding   a   number 
that  is  a  given  number  of  times  another  number. 

21.  The  Multiplicand  is  the  number  multiplied. 

22.  The  Multiplier  is  the  number  multiplied  by. 

23.  The  result  of  multiplication  is  called  the  Product. 

PRINCIPLES. —  The  multiplier  must  be  an  abstract  number. 
The  multiplicand  and  product  are  like  numbers.  The  prod- 
uct is  the  same  in  whatever  order  the  numbers  are  taken. 


DEFINITIONS.  8 

24.  Division  is  the  process  of  finding  how  many  times 
one  number  is  contained  in  another. 

25.  The  number  divided  is  the  Dividend. 

26.  The  number  by  which  the  dividend  is  divided  is  the 
Divisor. 

27.  The  result  of  division  is  the  Quotient. 

28.  When  the  divisor  is  not  exactly  contained  in  the 
dividend,   the    part    of    the    dividend    that    is   left   is  the 
Remainder. 

29.  PRINCIPLES.  —  The  remainder  and  dividend  are  like 
numbers.     When  the  divisor  is  abstract,  the  dividend  and 
quotient  are  like  numbers.     When  the  dividend  and  divisor 
are  concrete,  the  quotient  is  abstract. 

30.  The  Sign  of  Division  is  -r- ,  and  when  placed  between 
two  numbers  signifies  that  the  first  is  to  be  divided  by  the 
second. 

FACTORS   AND   MULTIPLES. 

31.  A  Factor  of  a  number  is  any  integer  that  will  exactly 
divide  it. 

32.  A  number  that  has  no  factors  except  itself  and  1  is 
a  Prime  Number. 

33.  A  number  that  has  other  factors  besides  itself  and  1 
is  a  Composite  Number. 

34.  A  prime  number  used  as  a  factor  is  a  Prime  Factor. 

35.  What  are  the  prime  factors  of  1155  ? 

Rule. —  Divide  the  number  by  any  prime  fac- 
tor that  will  exactly  divide  it.     Divide  the 
quotient    in    the   same    manner.       Continue 
the  division  until  a  quotient 
is  found   that   is    a    prime        5,  3,  7,  and  11 
number. 
The  divisors  and  the  last  quotient  are  the  primp  factors. 


4  SENIOR   ARITHMETIC. 

36.  Numbers  that  have  no  common  factor  or  divisor  are 
Prime  to  Each  Other. 

37.  A  Multiple  of  a  number  is  a  number  that  exactly 
contains  that  number.     15  is  a  multiple  of  5. 

38.  A  number  that  is  a  multiple  of  two  or  more  num- 
bers is  a  Common  Multiple  of  them.     24  is  a  common  multi- 
ple of  4  and  3. 

39.  The  least  multiple  of  two  or  more  numbers  is  their 
Least  Common  Multiple.     12  is  the  least  common  multiple  of 
3  and  4. 

40.  What  is  the  least  common  multiple  of  18,  27,  and 
30? 

Rule.  —  Divide  by  any  prime  number  that  2/18,  27,  30 
is  exactly  contained  in  two  or  more  of  3  /9,  27,  15 
the  numbers,  and  bring  doivn  the  quo-  3/3,  9,  5 
tient  and  undi-  1,  3,  5 

vided    numbers.       2x3  X  3  x  3  X  5  =  270.     Ans. 
Divide  again  as 

before,  continuing  th<>  dh'ision  until  the  quotients  and 
undivided  numbers  are  prime  to  each  otluT. 
The  product    of  the    divisors,   quotients,   and  undivided 
numbers  is  tJie  least  common  multiple. 

41.  A  number  that  is  a  factor  of  two  or  more  numbers 
is  a  Common  Divisor  of  them.     5  is  a  common  divisor  of 
30  and  40. 

42.  The  greatest  factor  of  two  or  more  numbers  is  the 
Greatest  Common  Divisor  of  them.     10  is  the  greatest  com- 
mon divisor  of  30  and  40. 

43.  'What  is  the  greatest  common  divisor   of  324   and 
372? 


Rule.  —  Divide     the     greater      324/372(JL_ 
number  by  the   less,   then 

the  divisor  by  the  remain-  48  /  324  (_6_ 

tier,  until  there  is  no  re-  tl_Ji 

mainder.     The  last  divisor  '       ^JR  — 

-is    the    greatest    common 
divisor. 

When  there  are  more  than  two  numbers,  first  find  the 
greatest  common  divisor  of  two  of  them,  then  of  this 
divisor,  and  a  third  number,  until  all  the  numbers  are 
used. 

CANCELLATION. 

44.  Cancellation  is  a  process  of  shortening  indicated  di- 
vision by  rejecting  the  same  factors  from  both  dividend 
and  divisor. 

45.  PRINCIPLES.  —  Rejecting  the  same  factor  from  divi- 
dend and  divisor  divides  both  by  that  factor. 

Dividing  both  dividend  and  divisor  by  the  same  number 
does  not  affect  the  quotient. 

COMMON   FRACTIONS. 

46.  A  Fraction  is  one  or  more  of  the  equal  parts  of  a 
unit.     The  unit  of  which  the  fraction  is  a  part  is  called 
the  Unit  of  the  Fraction,   and   one   of  the   equal   parts   is 
called  the   Fractional   Unit.     Two   or   more   fractions   hav- 
ing the  same  fractional  unit  are  Like  Fractions. 

47.  A  fraction  is  written  with  two  numbers,  one  above 
the  other,  with  a  line  between ;  as.  i. 

48.  The  number  below  the  line  in  a  fraction  is  the  De- 
nominator, and  shows  into  how  many  equal  parts  the  unit 
is  divided. 

49.  The  number  above  the  line  in  a  fraction  is  the  Nu- 
merator, and  shows  how  many  of  the  parts  are  taken. 


b  SENIOR   ARITHMETIC. 

50.  The  numerator  and  denominator  are  the  Terms  of  a 
Fraction. 

51.  A  Proper  Fraction  is  a  fraction  whose  value  is  less 
than  1.     Its  numerator  is  less  than  its  denominator;  as, 

i,  I- 

52.  An  Improper  Fraction  is   a  fraction  whose  value  is 
1  or  more  than  1.     Its  numerator  is  equal  to,  or  greater 
than,  its  denominator ;  as,  £,  |,  ^f . 

53.  An   integer  may  be  written  in  fractional  form  by 
giving  it  1  for  a  denominator ;  as,  5  =  f . 

54.  A  Mixed  Number  is  a  number  composed  of  an  integer 
and  a  fraction  ;  as,  Sj,  5|. 

55.  Reduction  of  Fractions   is  the  process   of   changing 
their  forms  without  changing  their  values. 

56.  PRINCIPLE.  —  Multiplying  or   dividing   both  terms 
of  a  fraction  by  the  same   number   does  not   change   the 
value  of  the  fraction. 

57.  To  reduce  a  fraction  to  Higher  Terms. 
Rule.  —  Multiply  both  terms  IHJ  the  same  number. 

58.  |  equals  how  many  60ths  ? 

NOTE.  — Since  the  new  denominator  must  be  60,  or  twelve  times 
the  given  denominator,  the  new  numerator  must  be  twelve  times  the 
given  numerator,  therefore  I  —  -jj-$. 

59.  To  reduce  a  fraction  to  its  Lowest  Terms. 

Rule. — Divide  both  terms  by  any  common  factor  ;  divide 
the  result  in  the  same  way  until  the  terms  are  prime 
to  each  other.  If  the  terms  are  large,  divide  by  their 
greatest  common  divisor. 

60.  To  reduce  a  Mixed  Number  to  an  Improper  Fraction. 
Rule.  —  Multiply  the  integer  by  the  denominator  of  the  frac- 
tion, add  the  numerator  to  the  prvduM,  and  write  the 
result  over  the  denominator. 


DEFINITIONS.  7 

61.  To  reduce  an  Improper  Fraction  to  an  Integer  or  Mixed 
Number. 

Rule.  —  Divide  the  numerator  by  the  denominator. 

62.  A  number  that  is  the  denominator  of  two  or  more 
fractions  is  the  Common  Denominator  of  those  fractions. 

63.  XOTE  1.  — The  common  denominator  of  two  or  more  frac- 
tions is  a  common  multiple  of  their  denominators.       . 

XOTE    2.  —  The  least  common  denominator  of  two  or  more  frac- 
tions is  the  least  common  multiple  of  their  denominators. 

64.  To  reduce  fractions  to  a  Common  Denominator. 

Rule.  —  Multiply  the  denominators  together  for  tin'  common 
denominator,  divide  it  by  the  denominator  of  each  frac- 
tion, and  multiply  both  terms  by  the  quotient. 

NOTE.  —  To  find  the  least  common  denominator,  find  the  least 
common  multiple  of  the  denominators,  and  proceed  as  above. 

ADDITION   OF   FRACTIONS. 

65.  Rule.  —  If  the  fractions  are  not  like  fractions,  reduce 
them  to  a  common  denominator,  add  their  numerators, 
and  place  the  sum  over  the  common  denominator.     Re- 
duce the  result  to  lowest   terms.     If  the  result  is  an 
improper  fraction,    reduce   it   to   an   integer  or   mixed 
number. 

To  add  mixed  numbers,   add    the    integers   and    fractions 

separately,  and  unite  the  results. 

SUBTRACTION   OF   FRACTIONS. 

66.  Rule.  —  If  the  fractions  are  not  like  fractions,  reduce 
them  to  a  common  denominator,  and  write  the  difference 
of  their  numerators  over  the  common  denominator. 

To  subtract  mixed  numbers,  subtract  integers  and  fractions 
separately. 


8  SENIOR    AK1THMKTIC. 

MULTIPLICATION   OF  FRACTIONS. 

67.    Rule.  —  Reduce  integers  and  ////,/w/  mi  miters  to  imj>r<>in't< 
fractions,  ami  multiply  tin-  HI' nu>>'<itn)-x  to'/ctlx'r  for  the 
numerator  of  the  product,  and  the  denominators  for  tin' 
denominator  of  the  product. 
Cancel  when  possible. 

68.  Two  ©r  more  fractions  joined  by  "  of  "  form  a  Com- 
pound Fraction.     The  word  "of"  is  equivalent  to  the  sign 
of  multiplication. 

DIVISION   OF  FRACTIONS. 

69.  PRINCIPLE.  —  To  divide  by  a  fraction  is  to  multiply 
by  that  fraction  inverted. 

70.    Rule.  —  Reduce  integers  and  mixed  number*  to  hi/proper 
fractions,  .and  multiply  the  dividend  by  the  divisor  in- 
verted. 
Cancel  when  possible. 

71.  $  -s-  £    may   be    written  —.     Such   an   expression   is 
called  a  Complex  Fraction,  and  ¥   is   used   simply   to    indi- 
cate division. 

72.  A  Complex  Fraction  is  a  fraction  having  a  fraction 
iii  one  or  both  of  its  terms. 


DECIMAL    FRACTIONS. 

73.  A  Power  is  the  product  of  equal  factors,  as  5  X  5  == 
25,  5  X  5  X  5  =  125.     25  is  the  second  power  of  5.     125 
is  the  third  power  of  5.      10  X  10  =  100.     10  x  10  X  10 
=  1000.     100  is  the  second  power  of  10.     1000  is  the  third 
power  of  10. 

74.  A   Decimal  Fraction  or  Decimal  is    a  fraction  whose 
denominator  is  10  or  a  power  of  10. 


DECIMAL   FRACTIONS.  9 

XOTE.  —  rr;iie  denominator  of  a  common  fraction  may  be  any 
number,  but  the  denominator  of  a  decimal  fraction  must  be  10,  100, 
or  1000,  etc. 

75.  A  decimal  is  written  at  the  right  of  a  period  ( . ) 
called  the  Decimal  Point. 

XOTE. —  It  is  not  customary  to  write  the  denominator  of  a  deci- 
mal. It  is  determined  by  the  position  of  the  decimal  point. 

76.  A  figure  at  the  right  of  a  decimal  point  is  called  a 
Decimal  Figure.     Tenths  are  written  like  dimes,  with  one 
decimal  figure.     Thus,  ^  =  .5.     Hundredths  are  written 
like  cents,  with  two  decimal  figures. 

Thus,  -T%%  =  .l>r> ;  Tfo  =  .07. 

Thousandths  are  written  like  mills,  with  three  decimal 
figures  ;  thus,  jfifr  =  .125 ;  Tifo  =  .016 ;  TTy%«  =  .004. 
Ten-thousandths  require  four  decimal  figures ;  hundred- 
thousandths,  five  ;  millionths,  six,  etc. 

77.  Xame  the  denominators  in  the  following :   .36 ;  .08  ; 
.294 ;  .1406  ;  .0001  ;  .263402. 

Change  to  decimals:  -^  ;  TVo5o  5  T o°o°<jVo  5  TiJcnyWo  5 
To%tf  5  TO^OO- 

78.  A    Mixed  Decimal  is  an  integer  and  a  decimal ;   as, 
16.04. 

79.  To  read  a  decimal. 

Rule. — Read  tin-  <l«-ini<tl  us  an   integer,  and  give  it  the 
<l<'noiniti«f'«>n  of  tlif  right-hand  figure. 

Read  the  following  numbers  : 


1.    .7 

6.    .16984 

11.    .500 

2.    .07 

7.    .10016 

12.    4.98625 

3.    .007 

8.    .0000054 

13.    38694.06 

4.    .700 

9.    :  W.  18006 

14.    9.98463004 

5.    .03065 

10.    .0005 

15.    235.850062 

10  SENIOR   ARITHMETIC. 

16.  100.000104       18.    3543.4536982       20.    303.303303 

17.  9.1632002         19.    30.3303303  21.    9.999999 

80.    To  write  a  decimal. 

Rule.  —  Write  the  numerator,  prefixing  ciphers  when 
necessary  to  express  the  denominator,  and  place  the 
point  at  the  left. 

NOTE.  — There  must  be  as  many  decimal  places  in  the  decimal  as 
there  are  ciphers  in  the  denominator. 

Express  decimally  : 

22.  Four   tenths.      Seventeen    hundredths.      Five   hun- 
dredths.     Three    hundred  twenty-five   thousandths..    Five 
thousandths.     Fifteen  thousandths.     Nineteen,  and  seven 
hundred  twenty-four  thousandths. 

23.  Seven  thousand  five  hundred  four  ten-thousandths. 
Sixteen,   and    125    ten-thousandths.     Six   ten-thousandths. 
Five  thousand  ten-thousandths. 

24.  Seventeen   thousand   two  hundred  eleven  hundred- 
thousandths.      Four    hundred-thousandths.      Fifteen   hun- 
dred-thousandths.      Eighteen,    and    two    hundred    sixteen 
hundred-thousandths.     One  hundred  twelve  hundred-thou- 
sandths. 

25.  Twenty-nine  hundredths.     Twenty -nine  thousandths. 
Twenty-nine  ten-thousandths.     Twenty-nine  hundred-thou- 
sandths.    One   and   one   tenth.     One   and   one   hundredth. 
One  and   one  thousandth.     One   and    one  ten-thousandth. 
One  and  one  hundred-thousandth. 

26.  324  and  one  hundred  twenty-six  millionth^.     4582 
and   36242    hundred-thousandths.       Seventeen   mil  Months. 
Five  hundred-thousandths.     Twenty-four,  and  three  thou- 
sand four  hundred  six  ten-millionths. 

27.  10  millionths.    824  ten-thousandths.    31  hundredths. 


DECIMAL   FRACTIONS.  11 

216  hundred-thousandths.     7846  hundred-millionths.    Four 
and  15  hundred-thousandths. 


28.  A  32'     ToWoU  36.      T30  40' 

29.  TW  33.    T$»JHi*iF       37.    Tin  41. 

30-  TVo%          34.     T^Wooo     38.    500tV        42. 

31-  iVoVo        35.    15To%o          39.     TT$fo        43. 

REDUCTION   OF  DECIMATE. 

81.  PRINCIPLES.  —  Ciphers  annexed  to  decimals  do  not 
change  their-  value. 

For  each  cipher  prefixed  to  a  decimal,  the  value  is  dimin- 
ished ten-fold. 

The  denominator  of  a  decimal  when  expressed  is  always 
1  with  as  many  ciphers  as  there  are  decimal  places  in  the 
decimal. 

82.  To  reduce  two  or  more  decimals*  to  a  Common  Denomi- 
nator. 

Rule.  —  Annex  ciphers  so  that  each  decimal  will  have  the 
same  number  of  decimal  figures. 

83.  Reduce  to  a  common  denominator  : 

44.  .5,  .017,  .1256,  .000155,  29.803. 

45.  .80062,  305.24,  70.5,  3.85263. 

46.  .1.  .0001.  1000.001,  1  .0100385. 

47.  .26,  .13682,  9.4,  25.,  8.63521. 

84.  Reduce  .375  to  a  common  fraction. 

.375  as  a  common  fraction  is  fVo5o-     This  in  lowest  terms 

=  1- 

Rule.  —  Write  the  numerator,  omitting  the  point.    Supply 
f//f  denominator,  <mrl  reduce  1<>  lowest  terms, 


12  SENIOR    ARITHMETIC. 

Reduce  to  common  fractions  : 


48. 

1.24 

53. 

.325 

58. 

16.144 

49. 

.16 

54. 

.113 

59. 

28.3695 

50. 

.325 

55. 

.7282 

60. 

34.000010 

51. 

.098 

56. 

2.25 

61. 

25.0000100 

52. 

.875 

57. 

.2425 

62. 

1084.0025 

85.    63.     Reduce  .371  to  a  common  fraction. 
a-i-fl..     An*. 


64.      .12J 
65.      .061 
66.      .621 

67.      .16§ 
68.     .33^ 
69.      .831 

70.     .87£ 
71.     .66f 
72.     .36| 

86.  To  reduce  a  common  fraction  to  a  Decimal. 

Reduce  f  to  a  decimal. 

|  =  3  times  J.  3  =  (3.0),  30  tenths.  J  of  3.0  =  (.7), 
7  tenths,  and  2  tenths  remainder.  2  tenths  =  20  hun- 
dredths.  |  of  .20  =  .05.  Hence  £  =  .7  +  .05  =  .75. 

Rule.  —  Annex  decimal  ciphers  to  the  numerator,  and  divide 
by  the  denominator.  Point  off  from  the  r'ujht  of  the 
quotient  as  many  places  as  there  are  ciphers  annexed. 

NOTES.  —  A  decimal  cipher  is  a  cipher  at  the  right  of  the  decimal 
point.  If  there  are  not  enough  figures  in  the  quotient,  prefix  ciphers. 
The  division  will  not  always  be  exact.  In  such  cases  write  the 
remainder  over  the  divisor  as  a  common  fraction,  or  place  the  sign 
+  after  the  decimal  to  show  that  the  result  is  incomplete.  Thus, 
\-=  .1425  or  .142+. 

87.  Reduce  to  decimals  : 

73.  $  77.  A  81.  I  85-  iV  89-  66it 

74.  |  78.  |  82.  o\  86.  ^5  90.  25.121 

75.  |  79.  |  83.  |  87.  12^  91.  161 

76.  §  80.  J  84.  T\  88.  33^  92.  16.25^ 


DECIMAL   FRACTIONS.  13 

ADDITION. 

88.    Add  .35,  4.375,  28.3&6S. 

Rule.  —  Write  the  numbers  so  that  decimal  points         .35 
stand  in  a  column.     Add  as  in  integers,  and        4.375  ^ 
place  the  point  in  the  sum  directly  under  the     z^r — * 
points  above. 
Find  the  sum : 

93.     24.36  94.       38.28006 

1.358  1.005 

.004  2.16 

1632.1  1873.148| 

96.  .175  +  1.75  +  17.5  +  175.  +  1750. 

97.  145.  +  14.5  +  1.45  +  .145  +  .0145. 

98.  32.58  +  28963.1  +  287.531  +  76398.9341. 

99.  1.  +  .1  +  .01  +  .001  +  100  + 10.  + 10.1  +  100.001. 

100.  1.923  +  .008  +  251.47  -f  1.961  +  0.0543  +  .006  + 
18.7. 

101.  Add  750.3521,  698.42001,  .005321,  3.5,  749.006984, 
36950.06,  875.942,  286.753. 

102.  Add  5  tenths ;  8063  millionths ;  25  hundred-thou- 
sandths ;    48    thousandths ;    17    millionths ;    95    ten-mil- 
lionths ;  5,  and  5  hundred-thousandths  ;  17  ten-thousandths. 

103.  Add  24J,  17^,  .0058,  7i,  9TV- 

SUBTRACTION. 

89.    Rule.  —  Write  the  numbers  so  that  the  decimal  point  of 

the  subtrahend  stands  directly  under  the  decimal  point 

in  the  minuend.      Subtract  as  in  integers,  and  place  the 

point  directly  under  the  points  above. 

NOTE.  —  It  is  sometimes  convenient  to  give  the  decimals  the  same 

denominator  by  annexing  ciphers. 


14  SENIOR    ARITHMETIC. 

104.    From  6.008     105.  38.          106.  26.34          107.  16.2600 
Take  3.154  .356  -  1.28983  1.0001 

108.  32.90596  -  75  114.  .00011  -  .000011 

109.  9.5  -  3.35006  115.  10  -  .1  +  .0001 

110.  856.2  -  8.562  lie.  8.75  +  .95  -f  .125 

111.  .1  —  .00001  117.  16  —  .00001  -f  27.69852 

112.  1000  -  .001  118.  2.5  —  .09  +  1.85  -  1.283 

113.  20  -  .00205  119.  83.1  -  8.31  +  .831 

120.  From  one  thousand  take  five  thousandths. 

121.  Take  17  hundred-thousandths  from  1.2. 

122.  From  8.5  take  eighty-four  hundredth  s. 

123.  Find  the  sum  of  500  thousandths  and  5  hundred- 
thousandths  and  from  it  subtract  y\. 

124.  From  17.37^  take  14.161. 

125.  Find  the  difference  between  T3^%  and  T§§ fo. 

126.  From  10  take  TV  ;  T^  ;  4.98  ;  1.05. 

127.  From  one  million  and  one  millionth  take  one  tenth. 

128.  From  1  tenth  take  1  millionth. 

129.  Which  is  the  greater  and  how  much,  one  tenth  or 
100  thousandths  ? 

130.  Prove  tha.t  ^  and  .500  are  equal. 

MULTIPLICATION . 

90.  Every  decimal  equals  a  corresponding  common  frac- 
tion, and  for  each  cipher  in  its  denominator  there  is  a  deci- 
mal figure  in  the  decimal  fraction. 

T$O  X  fV  =  Tinb-     (Three  ciphers  in  the  denominator.) 
.05  x  .3  =  .015.     (Three  decimal  places  in  the  decimal.) 
Rule.  — Multiply  as  in  integers,  and  give  to  the  product  as 
many  decimal  figures  as  there  are  in  both  multiplier 
and  multiplicand. 


DECIMAL    FRACTIONS.  15 

! 

NOTE.  —  If  there  are  not  figures  enough,  prefix  ciphers. 
Ciphers   at  the  right  of  a  decimal  have  no  value,  and  may  be 
omitted. 

Find  the  products : 

1.  .38  x  1.6.  11.  1.04  x  6£. 

2.  .015  X  .05.  12.  327f  X  4f . 

3.  7i  X  3.4.  13.  58.42  x  20.06. 

4.  50  X  .304.  14.  .0001  X  1000. 

5.  2.65  x  .104.  15.  .325  x  12i. 

6.  257  X  .354.  16.  .333  x  .333. 

7.  .296  X  124.  17.  .001542  X  .0052. 

8.  1.001x1.01.  18.  26x36.82. 

9.  13.33  X  1.3.  19.  2.84.  x  3J. 
10.  25.863  X  4i.  20.  11.11  X  100. 

91.  To  multiply  by  10, 100, 1000,  etc. 

21.  Multiply  1.265  by  100. 

B-emove  the  point  one  place  to  the  right  for        1.265 
each  cipher  in  the  multiplier.  100 

Do  not  write  the  multiplier.  126.500 

Oral. 

22.  3689.25  X  10.  27.  .5  X  100. 

23.  38.6422  x  100.  28.  .5  X  1000. 

24.  269.8342  X  1000.  29.  384.2  x  10. 

25.  100  X  23.85.  30.  .3659  X  100. 

26.  1000  X  1.52.  31.  .1000  X  .01. 

92.  To  multiply  by  200,  remove  the  point  to  the  right  and 
multiply  by  2. 

Oral. 

32.  86.44  X  200.  35.  750.5  X  5000. 

33.  3.894x3000.  36.  1.892x2000. 

34.  88.42  X  20.  37.  156.2  X  200. 


16  SUNIOU    A1MTHMKT10. 

93.  Written. 

38.  Find  the  product  of  1  thousand  by  one  thousandth. 
1  million  by  one  millionth. 

39.  Multiply  700  thousandths  by  7  hundred-thousandths. 

40.  Multiply  the  sum  of  2  millionths  and  10  thousandths 
by  their  difference. 

41.  Multiply  together  .35,  18.5,  28.004. 

DIVISION. 

94.  Since  in  multiplication  there   are  as   many  decimal 
places  in  the  product  as  in  both  multiplier  and  multipli- 
cand, in  division  the  quotient  must  have  as  many  places  as 
the  number  of  places  in  the  dividend  exceeds  those  in  the 
divisor. 

1.  Divide  12.685  by  .5. 

SOLUTION. — Since  there  are  three  decimal  places       .5/12.685 
in  the  dividend  and  one  in  the  divisor,  there  must  be  25  37 

two  in  the  quotient. 

Rule  I.  —  In  all  cases  divide  fs  in  integers,  then  place  the 
decimal  point. 

2.  Divide  399.552  by  192. 

Rule  II.  —  When  the  divisor  is  an  integer,  2.081 

place  the  point  in  the  quotient  directly  ooli 

over  the  point  in  the  dividend  in  long  ~T^^ 

division  (directly  under  in  short  divis-  1536 

ion).     Prove  by  multiplying  divisor  by  Jcj7> 

quotient.  192 

PRINCIPLE.  —  Multiplying  both  dividend  and  divisor  by 
the  same  number  does  not  change  the  quotient. 

3.  Divide  28.78884  bv  1.25. 


DECIMAL   FRACTIONS.  17 

Rule  III. —  When  the  divisor  contains  23.031  + 

(IfcItHftl  figures,  move  the  point  in   1-25  /28.78'884 
bnth  divisor  and  dividend  as  many 


orro 

to  the  right  as  there  are  deci-  ^ ° 

mal  places  in  the  divisor  (this,  in 


Ex.  3,  in  a  It  i plies  both  by  100),  then 

place  the  point  in  the  quotient  as  if  ~134 

the  divisor  were  an  integer.  ^25 

Q 

NOTE  1.  —  The  new  points  may  be  placed 
on  a  line  with  the  tops  of  the  figures,  and  the 
original  point  may  stand  to  preserve  the  reading  of  the  decimals. 

NOTE  2.  —  If  the  quotient  does  not  have  a  sufficient  number  of 
figures,  prefix  ciphers. 

NOTE  3.  — Before  commencing  to  divide,  see  that  there  are  at  least 
as  many  decimal  places  in  the  dividend  as  in  the  divisor. 
^     NOTE  4.  — If  there  is  a  remainder  after  all  the  figures  of  the  divi- 
dend are  used,  annex  decimal  ciphers  and  continue  the  division. 

NOTE  5.  —  It  is  not  usually  necessary  to  have  more  than  four 
decimal  figures  in  the  quotient. 

Find  the  quotients : 

1.  .288  4-  .64.  11.  315.432  -4-  .132. 

2.  .36  -4-  600.  12.  1.5906  -=-  241. 

3.  144  -4-  .12.  13.  36.25  -j-  1.25. 

4.  .25  -4-  .2500.  14.  75  -4-  .0125. 

5.  .12  -4-  30.  15.  125  -4-  .12J. 

6.  .96  -4-  .08.  16.  25  -4-  .25. 

7.  384.526  -*-  1.16.  17.  .25  -5-  25. 

8.  1440  -^  .0018.  18.  1000  -j-  .001. 

9.  1.225  -5-  4.9.  19.  .001  -*-  1000. 
10.  9.156  -4-  12.  20.  18.65  H-  100. 

95.    To  divide  by  10,  100,  1000,  etc.,  remove  the  point  one 
place  to  the  left  for  each  cipher  in  the  divisor. 


18  SKNIOll    ARITHMETIC. 

Oral. 

21.  38.64  -h  10.  25.  3.91  -h  1000. 

22.  .5  -5-  10.  26.  1.155  -f-  100. 

23.  558  -f-  100.  27.  398.42  -4-  1000. 

24.  1684.32  -f  1000.  28.  2.46  -*-  200. 

NOTE.  —  To  divide  by  200,  remove  the  point  to  the  left,  and  divide 
by  2. 

29.  386.54  -*-  2000.     31.  865.45  -5-  5000. 

30.  38.28  H-  400.       32.  2.5  -=-  500. 

PARTS  OF  100  OR  1OOO. 

96.  1.    What  part  of  100  is  12$  ?     25  ?     33J  ? 

2.  What  part  of  1000  is  125  ?     250  ?     333$  ? 

3.  How  much  is  1  of  100  ?     Of  1000  ? 

4.  How  much  is  |  of  100  ?     Of  1000  ? 

5.  Find  i  of  100.     Of  1000. 

6.  How  much  is  25  times  24  ? 

SOLUTION.  —  100  times  24  =  2400. 

25  times  24  =  i  as  much  as  100  times  24,  =  600. 

97.  To  multiply  by  25,  annex  two  ciphers,  and  take  1  of 
the  result. 

7.  Tell  how  to  multiply  by  33^ ;  by  12^  ;  by  250 ;  by 
125 ;  by  333£. 

Oral. 

8.  36  X  25.  11.   444  x  25.  14.    333J  X  30. 

9.  48  X  12i.  12.    320  X  33i.         15.    168  X  250. 
10.    24  X  33^.  13.    125  X  80.  16.    12$  X  48. 
17.    What  cost  650  oysters  at  50  cents  a  hundred  ? 

SOLUTION.  —  650  -f  100  =  6.50  hundred. 
$.50  x  6.50  =  ? 


DECIMAL   FRACTIONS.  19 

18.  What  will  be  the  cost  of  3850  laths  at  40  cents  a 
hundred  ? 

19.  What  is  the  freight  on  685  pounds  of  baggage  at 
$1.10  per  100  Ib. 

NOTK.  —  C.  means  100  ;  M.,  1000. 

20.  What  is  the  cost  of  4862  ft.  of  pine  lumber  at  $30 
per  M.  ? 

21.  Find  the  cost  of  38,586  bricks  at  $8.25  a  thousand. 

22.  What  will  583  heads  of  cabbage  cost  at  $3.50  a  hun- 
dred ? 

23.  At  $3.50  a  thousand,  what  will  be  the  cost  of  7800 
shingles  ? 

24.  At  $8.25  per  C.,  what  will  be  the  cost  of  2864  Ib.  of 
dried  fish  ? 

25.  At  $50  per  M.,  what  will  be  the  cost  of  3865  feet  of 
cherry  lumber  ? 

26.  What  is  the  cost  of  laying  5890  bricks  at  $9.00  a 
thousand  ? 

To  find  the  cost  of  merchandise  sold  by  the  ton,  divide 
the  price  by  2  and  proceed  as  above. 

27.  Three  loads  of  hay  weigh  7894  -lb.     What  will  the 
hay  bring  at  $12  a  ton  ? 

NOTE.  — 1000  lb.  will  cost  i  of  $12  =  $6.     $6  x  7.984  =  ? 

28.  What  cost  48986  lb.  of  railroad  iron  at  $35  a  ton  ? 

29.  Four  loads  of  coal  weigh  respectively  3896  lb.,  3524 
lb.,  4106  lb.,  and  3123  lb.     What  is  the  cost  of  the  coal  at 
$4.82  a  ton. 

ALIQUOT   PARTS   OF   $1.OO. 

98.    The   Aliquot  Parts   of    a   number   are    the    numbers 
which  are  exactly  contained  in  it. 

The  aliquot  parts  of  100  are  5,  20,  12£,  16 j-;,  33£,  etc. 


20  SENIOR   ARITHMETIC. 

99.    The  aliquot   parts  of   $  1,  commonly  used,  are   as 
follows  : 


6£  cents  =  $TV  25    cents  =  #J. 

8£  cents  =  $TV,.  33  J  cents  =  $J. 

12£  cents  =  $£.  50  cents  =  $£. 
16|  cents  ==  $£. 

1.    What  is  the  cost  of  69  books  at  16f  /  each  ? 

SOLUTION.  —60  books  will  cost  69  times  16f<',  or  69  x  $&  =  $-\9 
$11.50.     An*. 

100.  Oral, 

Multiply  : 

2.  33£  cents  by  36.  5.    25  cents  by  40. 

3.  12£  cents  by  24.  6.    .75  cents  by  4. 

4.  61  cents  by  32. 

7.  What  is  the  cost  of  : 

48  Ib.  of  bacon  at  12  £/  a  pound  ? 
80  hand  balls  at  50/  each  ? 
36  yd.  of  ribbon  at  33£/  a  yard  ? 
80  Ib.  of  candy  at  25/  a  pound  ? 

101.  Written. 

8.  Find  the  cost  of  the  following  : 

66  Ib.  of  pork  at  12  J/. 
148  Ib.  of  veal  at  16|^. 
48  boxes  of  strawberries  at  25  /'. 
48  Ib.  of  honey  at  25/. 
64  bars  of  soap  at  6i/. 
60  doz.  of  eggs  at  16f/. 
Find  the  cost  of  : 

9.  1580  Ib.  of  sugar  at  6^/  a  pound. 

10.  500  books  at  25/  each. 

11.  16  yd.  of  dress-goods  at  33^  a  yard. 


DECIMAL   FRACTIONS.  21 

12.  At  25/  a  pound,  how  many  pounds  of  butter  can  be 
bought  for  $8.00  ? 

SOLUTION.  —  As  many  pounds  as  25^  or  $|  is  contained  times  in 

$8  -i-  $i  =  8  x  f  =  32  Ibs.     Am. 

102.  Oral. 

Divide : 

13.  $5  by  33$?.  16.    $3  by  8^/. 

14.  $6  by  6J/.  17.    $4  by  25/. 

15.  $9  by  12i/.  18.    $4  by  66§X- 

19.  At  25^  each,  how  many  hats  can  be  bought  for  $6  ? 

20.  At  $1  a  pound,  how  many  pounds  of  cheese  can  be 
bought  for  $6  ? 

21.  At  33$  f  a  yard,  how  many  yards  of  linen  can  be 
bought  for  $  10  ? 

103.  Written. 

22.  At  75/  a  bushel,  how  many  bushels  of  barley  can  be 
bought  for  $125  ? 

23.  When  butter  is  25<?  a  pound,  how  many  pounds  can 
I  buy  for  $50  ? 

24-  How  many  dozen  eggs  at  16|  cents  a  dozen  can  be 
bought  for  $38? 

25.  At  12  i  cents  a  quart,  how  many  quarts  of  nuts  can 
be  bought  for  $10  ? 

REVIEW   OF   DECIMALS. 

104.  1.    Tell  how  to  locate  the  decimal   point  in  any 
sum.     In  any  remainder.     In  any  product.     In  any  quo- 
tient. 

2.    In  the  number  777.  what  is  the  local  value  of  the  7 
at  the  right  ?     The  second  7  ?     The  left-hand  7  ? 


22  SENIOR    ARITHMETIC. 

3.  Upon  what  does  the  value  of  any  figure  depend  ? 

4.  In  the  decimal  .777,  what  is  the  value  of  the  first  7 
at  the  right  ?     The  second  7  ?     The  third  7  ? 

5.  What  is  the  effect  of  removing  an   integral  figure 
one  place  to  the  right  ?     A  decimal  figure  ? 

6.  What  is  the  effect  of  removing  an  integral  figure 
one  place  to  the  left  ?     A  decimal  figure  ? 

Bead: 

7.  .0001,  .00196,  4.3, 
.0006,  .02789,  71.86, 
.0014,  .52000,  329.400, 
.0282,  .050798,  1.001, 
.5897,  .725386,  200.3278, 
.00001,  .500001.                 579000.00005, 
.00027,  .000829,  437.050609. 

Copy  and  write  decimally  : 

8.  1  tenth  ;  24  hundredths  ;  379  thousandths  ;  1000  ten- 
thousandths  ;  85  hundred-thousandths  ;  20079  millionths. 

9.  One  thousand  six  and  five  hundred  two  millionths. 

10.  Three  hundred  fifteen  thousand  one,  and  eleven  ten- 
thousandths  ;  thirty-eight,  and  seven  thousandths  ;  8  mil- 
lion 270  thousand  942,  and  5  thousandths  ;  seventeen  tenths. 

11.  Four  hundred  21,  and  5  ten-thousandths  ;  1  thousand 
27,   and  27  hundredths;  ninety-nine  and  ninety-nine  ten- 
millionths. 

Write  without  the  denominator  :. 


^O-O^J 


T(Oo  TO^O-O^  TO 


13.    Change  to  common  fractions  in  lowest  terms  : 
.028,  .0015,  .2175,  .000048,  .00075,  .45,  .8,  .75,  8.9375, 
91.16,  4001.  645,  9.156575, 


DECIMAL   FRACTIONS.  23 

Change  to  equivalent  decimals  : 

w.  i,  f,  A,  if.  A.  i,  2°if>  8,j,,  4Tv,  708H4- 

Change  to  common  fractions,  then  to  simple  decimals  : 

15.  .11,  .07|,  .18$,  .107|,  .121,  .08J,  .22|,  .045&,  .37|, 
.38f,  .54$,  .000051,  .78|,  .38J. 

Reduce  to  a  common  denominator  and  add : 

16.  50.06,    367.41,    200.200,    .12304,    40.0056,    7.5620, 
.096071. 

17.  1301.6,  904.02,  .547,  .0009,  .00001,  218.94,  203.410, 
1000,  .01. 

18.  100.101.     82.4,     401.009,     .00038,     60702,     10.10, 
574.68139. 

19.  5.628,   850.002,    9.00256,   37.0005,    724.6811,  3759, 
7000.0036,  2.25. 

20.  $11.78,  $347,  $5.06,  $218,  $20.07,  $42.0244,  $7.104, 
837.625. 

21.  4.76,   .390,.. 0915,    .00207,  841,   63.2,   .00234,   1.43, 
.00536. 

22.  .00908,  .0371,  24.5,  7.03,  .0127,  354,  .000781,  .0436, 
20.7354. 

Subtraction. 

23.  5.74  —  3.23  =  ?  26.    367.  —  1.52  =  ? 

24.  .876  —  .343  =  ?  27.    200  —  .02  =  ? 

25.  67.5  —  41.5  =  ? 

28.  Which  is  greater,  f  or  4  tenths  ? 

29.  How  much  more  is  $20  than  $17.84  ? 

30.  From  two  million  take  two  millionths. 

31.  T  bought  4  farms ;  one  contained  19.368  acres ;  one, 
27.96  acres;  one,  473.0008 .acres  ;  and  the  last  one,  73.7561 
acres.     I  sold  300.25  acres  ;  how  much  land  had  I  left  ? 

32.  From  1  inch  take  one  ten-thousandth  of  an  inch. 


24  SENIOR   ARITHMETIC. 

Multiply : 

33.  7.945  by  .3.  37.    7.853  by  23.16. 

34.  350  by  .42.  38.    1.36  X  20.04  =  ? 

35.  One  tenth  by  one  hundredth.     39.    27.27  X  4.0004  =  ? 

36.  25  units  by  25  tenths. 

40.  If  wheat  is  worth  $.38  a  bushel,  what  will   117.75 
bushels  cost  ? 

41.  Apples  sell  for  §1.28  a  bushel ;  how  much  money  will 
24  barrels  bring,  each  containing  2^  bu.  ? 

42.  Find  the  cost  of  3.325  Ib.  of  butter  at  18.75  cents  a 
pound. 

43.  What   will    6|  yd.    of  broadcloth  cost  at  $1.375  a 
yd.? 

44.  A  boy  paid  $.125  a  dozen  for  1.75  dozen  eggs ;  what 
did  they  cost  him  ? 

45.  3.64  x  .0002  x  1.756  x  4.004  =  ? 
Divide : 

46.  1738.89  by  .00417.  52.  42.475681  by  .29. 

47.  1237.6  by  26.  53.  40.20  by  .000012. 

48.  36.11  by  .021.  54.  $302.03  by  200. 

49.  2.38  by  .17.  55.  64.64006  by  .002. 

50.  36.82  by  .0003.  56.  12.9643  by  18.4. 

51.  437.96  by  2.8.  57.  759.806  by  90.3. 

58.  16£  +  3.06  -  |  +  .002  -  2.1  +  .03  -  j  +  .00£  =  ? 

59.  i  +  |  -  -65  +  .5  +  I  -  J  +  3.14  =  ? 

60.  Find    the    product    of    .003   multiplied    by  .06,  and 
divide  it  by  3. 

61.  A  certain  decimal  divided  by  1000  is  35.002.     What 
is  one  fifteenth  of  the  decimal  ? 


ACCOUNTS    AND    BILLS. 


25 


62.  The  sum  of  two  numbers  is  306.52 ;  one  of  them  is 
100.     What  is  the  other  ? 

63.  A  man  spent  $450,  which  was  .125  of  his  money. 
How  much  money  had  he  ? 

64.  Mr.  A.  bought  a  cow  for  $45,  which  was  .375  of  what 
he  paid  for  a  horse.     How  much  did  he  pay  for  the  horse  ? 

65.  John  spent  .75  of  his  money  for  a  book  and  had  $50 
left.     How  much  had  he  at  first  ? 


ACCOUNTS   AND   BILLS. 

105.  An  Account  is  a  record  of  indebtedness  for  articles 
bought  or  sold,  cash  paid  or  received,  or  services  rendered. 

106.  A  Debtor  is  a  person  who  owes  a  debt. 

107.  A  Creditor  is  a  person  to  whom  a  debt  is  owed. 

108.  A  Bill  is  a  written  statement  of  a  debtor's  account, 
made  by  the  creditor. 

109.  A  Receipt  is  a  creditor's  written  acknowledgment 
that  he  has  received  payment  of  part  or  all  of  a  debt. 

110.  A  bill  is  receipted  when  its  payment  is  acknowl- 
edged in  writing,  by  the  creditor,  or  by  some  authorized 
person. 

NOTE.  —  The  sign  @  is  for  at.     Dr.  is  for  debtor,  and  Cr.  for 
creditor. 


JAMES  P.  BARNES, 


BILL  FORMS. 

SYRACUSE,  tf.  Y.,  July  1,  1896. 
Bought  of  BEY  BROS.  &  Co. 


50  yd.  Brussels  Carpet      @ 

$1 

15 

I 

24    «     Oil  Cloth 

35 

4  doz.  pair  Merino  Hose    " 

3 

50 

2  Willow  Chairs                  " 

4 

50 

$ 

SENIOR    ARITHMETIC. 

RECEIPTED  BILL  WITH  CREDITS. 

ROCHESTER,  N.  Y.,  Jan.  2,  1896. 


MRS.  JOHN  F.  WHITE, 


To  BURKE  &  WHITE,  Dr. 


Nov. 

6 

4  Ib.  Coffee                 @ 

$ 

27 

$ 

« 

6 

28  Ib.  Sugar                  « 

sy2 

« 

18 

5  gal.  Molasses           " 

60 

Dec. 

11 

18  Ib.  Rice                     " 

7X 

it 

15 

2  bbl.  Potatoes             " 

1 

80 

tt 

19 

28  Ib.  Butter                  " 

21 

CR. 

Nov. 

18 

Cash 

3 

50 

Dec. 

28 

u 

4 

75 

Balance  due, 

Received  payment,  Jan.  15,  1896, 

BURKE  &  WHITE, 

By  JOHN  R.  PIERCE. 


FORM  OF  A  RECEIPTED  BILL. 

NEW  YORK,  June  30,  1896. 
JEROME  A.  PHELPS, 

In  account  ivith  D.  0.  POTTER  &  Co. 


May 

14 

12  bbl.  Flour                     @ 

$6 

50 

$ 

it 

14 

6  tubs  Butter,  684  Ibs.        " 

24 

June 

10 

5  bbl.  Beef                          " 

25 

28 

« 

25 

450  Ib.  Ham                       « 

9% 

Received  payment,  . 

D.  O.  POTTER  &  Co. 


INDICATED   OPERATIONS.  27 


INDICATED   OPERATIONS. 

111.  The  Parenthesis,  (  ),  indicates  that  all  the  numbers 
contained  therein  are  to  be  taken  together. 

112.  Brackets,  [  ],  Braces,  {  },  and  the  Vinculum,  ~~     ~~, 
have  the  same  use  as  the  parenthesis. 

113.  When  the  parenthesis  is  not  used,  operations  indi- 
cated by  X  or  -f-  must  be  performed  first.     Thus, 

1.  12  -.-4x2+  36  -j-4-2x4  =  ? 

SOLUTION.  — 

12  -f  4  x  2  =  6. 

36  -f-  4  =  9.  6+9-8  =  7.     Ans. 

2x4=8. 

2.  4  +  3x2  =  ?  5.   4  x  (3  +  2)  =  ? 

3.  (4  +  3)  X  2  =  ?  6.    8  +  4  -r-  2  =  ? 

4.  4x3  +  2  =  ?  7.  (8  +  4)  H-  2  =  ? 

NOTE.  —  When    one    parenthesis,   brace,   or  vinculum   includes 
another,  first  remove  the  inner  one. 

114.  Find  the  value  of  : 

1.  15  +  3  X  6  +  10  -*-  5. 

2.  (6  +  4)  x  (3  +  2)  -  (8  X  5). 

3.  18  ^3x2  +  8x2-^4- 6. 

4.  2  +  12  -^  4  -  (10  +  6  -f-  4)  -^  3. 

5.  (11  +  4)  -s-  3  +  6  X  4. 

6.  3  +  4  x  6  -f-  (15  +  9  +  3). 

7.  164  +  16  -  250  -r-  10  +  16  X  3. 

8.  17  +  3x4x6  +  3-f-3  +  3. 

9.  [39  +  8  ^-  2  +  7]  X  6. 

10.    [6  +  15  x  3  -  (6  +  16  -v-  8  +  4)]  -f-  8  +  5. 


28  SENIOR    ARITHMETIC. 


MISCELLANEOUS. 

115.  1.  I  have  four  pieces  of  broadcloth.  The  first  con- 
tains 13.7642  yd. ;  the  second,  22.008  yd. ;  the  third,  15.027 
yd. ;  and  the  fourth,  19.255  yd.  How  many  yards  in  all  ? 

vX&    From  a  piece  of  ribbon  containing  103f  yd.,  73|  yd. 
were  sold.     How  many  yards  were  left  ? 

3.  How  many  yards  of -muslin  at  $.12^  a  yard  will  it 
take  for  4  pair  of  curtains,  if  each  curtain  contains  3.375 

yd.? 

4.  I  have  14.735  yd.  of  lace,  and  desire  to  cut  it  into 
seven  equal  strips ;  how  much  will  there  be  in  each  strip  ? 

5.  What  will  be  the  cost  of  a  hat  at  $7.50,  a  pair  of 
gloves  at  $1.13,  a  veil  at  $1.25,  and  a  parasol  at  $3.375  ? 

6.  Arrange  the  following  articles  in  the  form  of  a  bill : 
7  qt.  of  molasses  at  $.15  a  qt.,  3  pk.  of  apples  at  $1.28  a 
bushel,  30  Ib.  of  sugar  at  $.08^  a  pound,  and  12  bu.  of 
potatoes  at  $.29  a  bushel. 

y  7.  A  grocer  bought  three  bunches  of  bananas  at  $1.54 
a  bunch.  The  first  bunch  contained  73  bananas,  the  second 
54,  and  the  third  97.  He  sold  them  all  at  30/  a  dozen ; 
did  he  gain  or  lose,  and  how  much  ? 

8.  The  first  year  in  business  a  grocer  made  $2374.68, 
the  second  $1529.47,  and  in  the  third  year  he  lost  $300. 
His  expense  each  year  averaged  $928.45  ;  how  much  money 
had  he  gained  at  the  end  of  three  years  ? 

9.  What  will  9  barrels  of  flour  cost,  if  28  barrels  cost 
$173.60  ? 

10.  I  bought  437  heads  of  lettuce  at  $5  a  hundred,  and 
sold  them  at  $.08  a  head ;  what  was  my  gain  ? 


MISCELLANEOUS.  29 

Find  the  cost  of : 

11.  6824  Ib.  of  coal  at  $4.68  a  ton. 

12.  2384  Ib.  of  coal  at  $5.67  a  ton. 

i/is!  8972  ft.  of  lumber  at  $35.40  a  thousand. 

14.  6854  Ib.  of  hay  at  $16.50  a  ton. 

15.  4836  bricks  at  $9.45  per  M. 

16.  895  ft.  of  lumber  at  $19.75  per  M. 

17.  What  part  of  4.50  is  3.33^  ? 

18.  What  part  of  3.625  is  1.5  ? 
/i9.  What  part  of  6.2  is  3.25  ? 

20.  1.1  is  what  part  of  7.4  ? 

21.  A  father  left  his  son  $24000,  which  was  .375  of  his 
estate.     What  was  the  value  of  the  estate  ? 

22.  Divide  26  by  2J,  and  multiply  the  result  by  17.345. 

23.  Divide  f  of  .375  by  f  of  f  of  .298. 

24.  The  product  of  three  numbers  is  167.7.     Two  of  the 
numbers  are  3.25  and  5.16.     What  is  the  other  ? 

25.  What  number  divided  by  2.86  equals  .34  ? 

26.  What  number  diminished  by  38.64  leaves  .356  ? 

27.  A  man  bought  8.5  yd.  of  cloth  at  $3.33^  a  yard, 
12.4  yd.  at  $2.75, 18J  yd.  at  $4.375,  and  24f  yd.  at  $2.875. 
How  many  bushels  of  corn  at  43f  cents  a  bushel  will  pay 
for  the  cloth  ? 

28.  .5  of  a  number  exceeds  .45  of  it  by  20.    What  is  the 
number  ? 

SOLUTION:   .5  — .45  =.05.      Now  the  question  is,  20  is  .05   of 
what  ?     20  -f  .05  =  400. 

»    29.    At  85^  a  yard,  how  many  yards  of  cloth  can  be  pur- 
chased for  $29.75. 


30  SENIOR   ARITHMETIC. 

30.  Divide  $785  among  A,  B,  and  C,  so  that  C  will  have 
$185  more  than  each  of  the  others. 

31.  JL__  _^?046_  _=? 
.05        .4  x  .005  +  .002  x  .125 

32.  What  part  of  .876  is  .31536  ? 

33.  If  .375  of  a  ton  of  coal  cost  $1.25,  what  will  7.125 
tons  cost  ? 

34.  What  is  .3  of  a  number  when  .8  of  it  is  80  ? 

35.  How  many  thousandths  in  3  units  ? 

36.  How  many  thousandths  in  .1  ? 

37.  Express  *  of  one  hundredth  as  a  decimal. 

^38.    The  salary  of  the  President  of  the  U.  S.  is  $50,000 
a  year.     How  much  does  he  receive  per  day  ? 

116.   Review  of  Factors,  Multiples,  Divisors,  and  Cancellation. 

1.  Define  factor,  composite  number,  prime  number,  and 
prime  factor. 

2.  Find  the  prime  factors  of  5075;  of  9576;  of  3150; 
of  6006. 

3.  Find  the  sum  of  the  prime  factors  of  34650. 

4.  Find  the  prime  factors  of  2310  ;  of  17199  ;  of  6840. 

5.  81158  is  the  product  of  what  prime  factors  ? 

6.  Find  the  largest  prime  factor  of  12600. 

7.  What  is  a  common  divisor  of  two  or  more  numbers  ? 

8.  What  is  the  greatest  common  divisor  of  two  or  more 
numbers  ? 

9.  When  are  numbers  prime  to  each  other. 
Find  the  greatest  common  divisor  of : 

/ID.    672  and  960.  13.    1650  and  1920. 

11.  616  and  1012.  14.    696,  1218,  and  1160. 

12.  272  and  428.  15.   450,  720,  and  810. 


MISCELLANEOUS.  31 

16.  What  is  the  greatest  prime  factor  common  to  4242 
and  2626. 

17.  A  grocer  had  84  bananas  and  126  lemons,  which  he 
wished  to  put  into  bags,  each  bag  containing  the  largest, 
number  possible,  and  each  containing  the  same  number. 
How  many  could  be  put  into  each  bag  ? 

18.  A  man  has  three  fields  containing  respectively  14, 
18,  and  22  acres.     He  wishes  to  cut  them  into  the  largest 
possible  lots  of  equal  size.     How  much  land  will  each  lot 
contain  ?     How  many  lots  will  each  field  contain  ? 

19.  What  is  a  multiple  of  a  number  ?    A  common  multi- 
ple ?     The  least  common  multiple  ? 

Find  the  least  common  multiple  of : 
^.    96,  196,  42,  and  54.  23.    252,  462,  and  1092. 

21.  45,  36,  70,  and  90.  24.    120,  280,  and  308. 

22.  36,  40,  42,  and  48.  25.    36,  110,  98,  and  66. 

26.  Find  the  least  common  multiple  of  the  even  num- 
bers to  and  including  20. 

27.  What   is  the  least  sum  with  which  I  can  buy  an 
exact  number  of  chairs  at  $6,  $8,  or  $5  each  ? 

28.  What  is  the  smallest  sum  of  money  that  may  be 
expended  by  using  an   exact  number   of   nickels,   dimes, 
quarters,  or  3-cent  pieces  ? 

How  many  pieces  of  each  kind  will  the  sum  contain  ? 

r  29.  John  can  run  around  a  block  in  6  minutes,  James  in 
8  minutes,  and  Henry  in  9  minutes.  If  they  start  to- 
gether, how  long  before  they  will  all  be  together  again  at 
the  starting-point  ? 

30.  What  is  the  shortest  piece  of  rope  that  can  be  cut 
into  pieces  32,  36,  and  44  feet  long  ? 

31.  WThat  is  cancellation  ? 


32  SENIOR    ARITHMETIC. 

32.    Of  what  use  is  cancellation  ? 

Find  results  of  the  following  by  cancellation  : 

33        18  X  36  X  48  36  28  x  56  x  30 

24  X  6  X  12  14  x  3  X  5 

10  X  6  X  4  37  28  X  32  x  7 

4x6  14  X  35  X  2 

35.        18  X  9  X  10  X  5  3g  34  x  9  X  5 


6x8x2  25x17x3 

39.  240  X  48  X  70  X  18  -v-  42  x  15  X  54  x  7  =  ? 

40.  Divide  the  product  of  25,  14,  and  11  by  the  product 
of  15,  7,  and  22. 

41.  How  many  bushels  of  wheat  at  $1.10  a  bushel  must 
be  given  for  6  pieces  of  cloth  each  containing  33  yards  at 
50  cents  a  yard  ? 

42.  How  many  cords  of  wood  at  $3  a  cord  will  pay  for 
30  Ib.  of  sugar  at  5  cents  a  pound  ? 

43.  If  8  men  can  do  a  piece  of  work  in  6  days,  in  how 
many  days  can  12  men  do  it  ? 

Y/   44.    How  many  pounds  of  sugar  can  be  bought  for  $7 
if  21  Ib.  cost  $1.05  ? 

45.  How  many  pounds  of  maple   sugar  at  12  cents  a 
pound  must  a  farmer  exchange  for  15  pounds  of  coffee  at 
24/  a  pound  ? 

46.  A  milkman  exchanges  8  cans  of  milk,  30  quarts  in 
a  can,  at  4  cents  a  quart,  for  3  pieces  of  sheeting,  40  yards 
in  a  piece.     What  is  the  price  of  the  sheeting  per  yard  ? 

MISCELLANEOUS  RE  VIE  W  OF  COMMON  FRA  CTIONS. 

117.    Oral. 

Y    1.    Add  \  and  | ;  ^  and  1 ;  f  and  f  ;  f  and  | ;  £  and  $ ; 
|  and  1^. 

2-    i-i  =  ?     i-J  = 


MISCELLANEOUS   KEVIEW   OF   COMMON   FRACTIONS.     33 

3.  Reduce  to  improper  fractions  3i,  T|,  8S|,  5*,  7f,  8|, 
16|,  lof. 

4.  Reduce  to  integers  or  mixed  numbers  J?5-,  -1/-,  -y5-,  -y, 

¥->  ¥,  ft,  -W-,  ¥2°- 

u/-l    Multiply  16  by  §  ;  45  by  f  ;  18  by  |  ;  45  by  ft  5  §  by 
9;   |  by  32;  ±*  by  16  ;  J  by  27. 

6.  Find  product  of  :   f  X  f  ;  T9T  X  ^  ;   3J  X  U  ;    J  X 
i?5  f  X  -V;  41X6J. 

7.  Find  |  of  24  ;  f  of  12  ;  £  of  30  ;  |  of  27  ;  f  of  45  ; 
I  of  40. 

8.  Find  l  of  i  ;    i  of  f  ;   f  of  j  ;  ^  of  f  ;  |  of  j«  ;  J  of 

2J.. 

9.  Divide  f  by  3;  f  by  4;  ^  by  12;  TeT  by  11;  4J  by 
3;  i  by  6;  4§by6;  10  by  $  ;  8  by  £  ;  «  by  8  ;  t\  by  22  ; 
f  by.  9. 

\xm    Divide  4  by  J  ;  8  by  J  ;  9  by  -|  ;  16  by  f  ;  24  by  f  ; 
13  by  |;  11  by  f;  12  by  If. 

11.  Divide  i  by  |  ;   «  by  |  ;   J  by  J  ;    f  by  |  ;   ^  by  f  ; 
f  by  f  ;  5i  by  2J. 

12.  Divide  1  by  : 


)  1)   8)  T^J   T> 


NOTE.  —  1  divided  by  a  fraction  equals  that  fraction  inverted. 

13.  |  of  12  =  ?    9  =  what  part  of  12  ?    9  is  |  of  what  ? 

14.  3X?  =  §;    f  •*•?=*  iJ   iX?  =  J;   (f  +  r-ij  ) 

2  =  i  ;  |  -  ?  =  f  . 


6.    What  part  of 

6  is  4  ?  J  is  J  ?  |  is  §  ?          5£  is  2\  ? 

11  is  5?  j  is  K          j  is  J?  |  is    i? 

17.   9  is  |  of  what  ?     5  is  f  of  what  ?  6  is  f  of  what  ? 


34  SENIOR   ARITHMETIC. 

18.  Change  |  to  24ths ;  f  to  loths ;  f    to  32nds ;   £    to 
20ths. 

19.  A  man  owning  |   of  a  farm,  sold  ^   of  his  share. 
What  part  did  he  sell  ?     How  much  remains  ? 

20.  At  12i/  a  dozen,  how  many  dozen  of  eggs   can   I 
buy  for  $3  ? 

^r  *• 

21. /At  6J/  a  box,  how  much  will  8  boxes  of  berries  cost  ? 
v/22.    John  has  56  cents,  and  James   \   as  much.     How 
much  have  both? 

23.  A  can  do  a  piece  of  work  in  4  days ;  B  can  do  the 
same  piece  of  work  in  2  days.     What  part  of  the  work  can 
each  do  in  a  day  ? 

24.  A  can  mow  a  field  in  3  days,  and  B  in  4  days.    What 
part  of  the  field  can  they  mow  in  a  day  if  both  work 
together  ? 

A  can  mow  \  of  it  in  1  day,  and  B  can  mow  \  of  it  in  1  day. 
Both  working  together  can  mow  the  sum  of  ^  and  £  =  ^  of  it  in  1 
day. 

25.  C  can  do  a  piece  of  work  in  2  days  and  D  can  do  it 
in  4  days.     In  what  time  can  they  both  do  it,  working 
together  ? 

C  does  \  of  it  in  1  day,  and  D  i  of  it  in  1  day.  Therefore  both 
can  do  i  +  |  =  £  of  it  in  1  day.  Since  both  can  do  f  of  it  in  1  day, 
it  will  take  as  many  days  to  do  |  or  the  whole  of  it,  as  f  is  contained 
times  in  -|,  or  \\  days.  .4ns. 

NOTE.  —  f  divided  by  f  gives  the  same  result  as  4  divided  by  3. 

26.  \  of  my  money  is  gold,  and  \  as  much  is   silver. 
What  part  of  my  money  is  silver  ? 

/27.    If  a  boy  can  earn  $2^  in  1  week,  how  much  can  3 
boys  earn  in  4  weeks  ? 

28.  James  sold  a  book  for  28  cents,  which  was  §  of  what 
it  cost  him.  What  did  it  cost  him  ? 


•• 

•' 


MISCELLANEOUS    REVIEW    OF    COMM.6X    FRACTIONS.     35 

.      .• 

29.  The  difference  between  £  of  a  number  and  J  of  it  is 
6.     What  is  the  number  ? 

SOLUTION.  —  The  difference  between  \  and  \  is  i.  Now  the  ques- 
tion is  6  is  |  of  what  ? 

30.  A  boy  12  years  of  age  is  \  as  old  as  his  father.    How 
old  is  his  father  ? 

Written. 

i/%\..  A  farmer  having  1200  bushels  of  potatoes,  sold  \  of 
them  at  one  time,  \  at  another,  and  350  bushels  at  another. 
How  many  bushels  had  he  left  ? 

32.  A  mechanic  whose  wages  are  $5  per  day  uses  -fV  of 
his  weekly  earnings  for  board,  arid  f  for  clothing  and  other 
expenses.     How  many  dollars  does  he  save  weekly  ? 

33.  Which  is  greater  and  how  much,  \\  or  \\  ? 

34.  If  it  takes  27  days  to  do  a  piece  of  work,  how  long 
will  it  take  to  do  f  of  it  ? 

35.  If  a  horse  is  worth  $100,  and  a  cow  is  worth  f  as 
much  as  the  horse,  what  is  the  cow  worth  ? 

36.  John  has  in  the  bank  $45  and  draws  out  f  of  it. 
How  much  remains  in  the  bank  ? 

37/  What  will  16  pair  of  shoes  cost  at  f  3|  a  pair  ? 
v'aS.    If  a  farmer  has  23  sheep  and  sells  them  at  $3T9<y 
apiece,  how  much  does  he  receive  for  the  sheep  ? 

39.  What  is  the  cost  of  ^  of  a  pound  of  cheese  at  10/  a 
pound  ? 

40.  What  is  the  cost  of  f  of  a  yard  of  cloth  at  $li  a  yard  ? 

41.  f  X  f  X  6^  X  T\  X  3  X  if  =  ? 

42.  What  is  the  value  of  3J  of  8|  of  f  of  if  ? 

43".  A  man  sold  3^  tons  of  hay  at  one  time,  7f  at  another, 
and  enough  the  third  time  to  make  20  tons.  How  many 
tons  did  he  sell  the  third  time  ? 


36  SENIOR    ARITHMETIC. 

44.  T^  plus  ^  plus  ^  plus  ^  and  how  many  more  will 
make  3  ? 

45.  A  man  having  a  farm  of  96  acres,  sold  ^  of  an  acre 
to  one  man,  1  of  an  acre  to  another,  f  of  an  acre  to  another, 
and  j1^  of  an  acre  to  another.     How  many  acres  had  he 
left?/ 

\  46.  If  two  men  were  90  miles  apart  and  each  should 
travel  23|  miles  toward  the  other,  how  many  miles  would 
they  then  be  apart  ? 

47.  If  George  has  ^  of  a  dollar  and  \  of  a  dollar,  and 
Henry  has  £  of  a  dollar  and  ^  of  a  dollar,  which  has  the 
greater  amount  and  how  much  ? 

48.  A  man  bought  3  loads  of  wood  containing  respect- 
ively li  cords,  If  cords,  and  If  cords.     How  many  cords 
of  wood  did  he  buy  ? 

49.  I  paid  $10^  for  hay,   $15f  for  coal,   and  $6|  for 
wood.     What  did  I  pay  for  all  ? 

50.  Mr.  Jones  paid  $525^  for  a  span  of  horses,  and  sold 
them  for  $625f .     How  much  did  he  gain  ? 

1/51.  L.  W.  and  J.  E.  Council  paid  $4500 1  for  a  store  and 
its  contents.  They  sold  it  for  $5025|.  How  much  did 
they  gain  by  the  operation  ? 

52.  A,  B,  C,  D,  and  E  own  respectively  ^,  |,  },  T9<y,  and 
\\  acres  of  land.     How  much  do  they  all  own  ? 

53.  A  gentleman  having  $1700,  paid  $825|  for  horses, 
$230f  for  cows,   {fl>150|  for  oxen,  and   $407f   for  sheep. 
How  much  money  had  he  left  ? 

54.  Mr.    Blanchard  paid  $8T9Q-  for  shovelling  his  walk, 
$5f  for  trimming  his  grape-vines,  and  $6f  for  sifting  his 
ashes.     He  gave  the  man  a  20-dollar  bill  and  a  c  ^lar  bill. 
How  much  money  should  Mr.  B.  receive  in  retun 


MISCELLANEOUS    REVIEW    OF   COMMON    FRACTIONS.     37 

55.  If  I  add  2  to  each  term  of  the  fraction  J,  will  its 
value  be  increased  or  diminished,  and  how  much  ? 

56.  Mr.  Homer  has  10i  acres  of  wheat,  6|  acres  of  corn, 
20|  acres  of  barley,  and   16|   acres   of  rye.     How  many 
acres  of  grain  has  he  ? 

•^57.    What  is  the  quotient  of  389  divided  by  1556,  ex- 
pressed in  its  simplest  form  ? 

•58.      -Ji±J)  -M        =  ? 

f  of  13  of  ft  of  3 

59.  816  is  I  of  what  number  ? 

60.  From  §  of  }§  take  f . 

61.  The  product  of  two  factors  is  10^ ;  one  factor  is  3|. 
What  is  the  other  ? 

62.  |  +  6f  +  9|  +  A  +  ^  =  ? 

63.  (T\  of  2i  of  T37)  x  (f  of  3i  of  8  of  J)  =  ? 

64.  The  sum  of  two  numbers  is  19|^.     One  of  the  num- 
bers is  12f.     What  is  the  other  ? 

65.  Reduce  -£  to  its  lowest  terms. 

66-    (I  -  I)  X  (|  +  |)  =  ? 
67.    Change  to  simple  fractions  : 

il     -H     i_ofj         7^         |_ofj 

9  '  61'      16     '  Jof2i'  Jof 

A  can  do  a  piece  of  work  in  4  days,  B  can  do  the 
same  work  in  5  days,  and  C  in  6  days.  In  what  time  can 
all  do  it  together  ? 

69.  A  tank  has  3  supply  pipes.  It  can  be  filled  in  6 
hours  by  the  first  pipe,  in  7  hours  by  the  second,  and  in  8 
hours  by  the  third.  In  how  many  hours  can  the  tank  be 
filled  by  the  three  pipes  together  ? 


38  SENIOR   ARITHMETIC. 

70.  A  and  B  can  do  a  piece  of  work  in  3  days.     A  can 
do  it  alone  in  5|  days.     In  what  time  can  B  do  it  alone  ? 

SOLUTION.  —  Both  can  do  |  of  it  in  1  day.  A,  alone,  can  do  fi  of 
it  in  1  day.  3  —  i2r  —  /35  the  part  A  can  do  in  1  day.  Since  he  can 
do  35j  of  it  in  1  day,  he  can  do  -jj-jj,  or  the  whole  of  it,  in  as  many  days 
as  /3  is  contained  times  in  ||,  or  33  -=-  5  =  6|  days. 

71.  £  of  my  property  is  invested  in  land,  f  of  the  re- 
mainder in  business,   and   f   of   the  remainder,  which  is 
$2400,  is  in  the  bank.     How  much  property  have  I  ? 

72.  What  is  the  value  of  f  1§  -j-  6  —  §  of  £  -f  i 


/  73.    A  farmer  sold  11  doz.  eggs  at  14^/  a  dozen,  and  took 
his  pay  in  sugar  at  5£/  a  pound.    How  much  did  he  receive  ? 


74.  Find  the  value  of  —$.  --  L  i  of  1  -s-  1. 

1  °f  t 

75.  A  boy  having  spent  1  of  f  of  his  money  for  a  knife, 
had  $2.25  left.     How  much  did  he  pay  for  the  knife  ? 

76.  A  father  left  $39000  to  his  two  children,  dividing  it 
so  that  the  daughter  received  |  as  much  as  the  son.    What 
was  the  share  of  each  ? 

77.  A  person  owning  f  of  a  steamboat,  sold  j  of  iiis  share 
for  $17360.     What  was  the  value  of  the  boat  ? 

78.  After  spending  ^  of  my  money  and  ^  of  the  remain- 
der, I  had  $300  left.     How  much  had  I  at  first  ? 

79.  If  i  of  |  of  a  bushel  of  apples  cost  f  of  T9^  of  a  dol- 
lar, what  will  §  of  |  of  a  bushel  cost  ? 

80.  How  many  pounds  of  honey  at  \  of  ^  of  a  dollar  a 
pound  can  be  bought  for  f  of  2|  dollars  ? 


81.    Simplify 

*  J  of 

82    ?  of  >  +  *  °f 

12      '  2  x  3§ 


QUESTIONS.  39 

83.  Divide  the  product  of  5  times  ?1  plus  L^Lt  by  5i  . 

4$  *          15       J  4i 

84.  Divide  f  of  -5  by  f  of  -i  . 

Find  the  cost  of  the  following : 

85.  315i  Ib.  of  tea  at  $.37£  a  pound. 
34|  Ib.  of  coffee  at  $.18f  a  pound. 
3105  j  Ib.  of  pork  at  $.121  a  pound. 
30691  bu.  of  wheat  at  $1.12i  a  bushel. 
36J  doz.  of  eggs  at  $.12J  a  dozen. 

26|  yd.  of  sheeting  at  $.07f  a  yard. 

86.  A  owns  f  of  a  farm  and  B  owns  the  remainder ;  f  of 
the  difference  of  their  shares  is  worth  $2400.    What  is  the 
value  of  the  farm  ? 

87.  Divide  $3 J  among  some  poor  children,  giving  each  i 
of  a  dollar.     What  will  be  the  number  of  children  ? 

y/88.    Two  men  hire  a  pasture  for  $25.    A  puts  in  8  horses 
and  B  12  horses.     How  much  should  each  pay  ? 

89.  Add  8  to  both  terms  of  the  fraction  £,  and  find  how 
muptf  you  have  increased  or  diminished  it. 

r  90.    Subtract  4  from  each  term  of  the  fraction  f  and  find 
how  much  it  has  been  increased  or  diminished. 

QUESTIONS. 

118.    1.    What  is  a  decimal  ?     How  are   decimals  writ- 
ten ?     Why  are  they  called  decimals  ? 

2.  How  many  decimal  places  are  needed  to  write  ten- 
thousandths  ?     Millionths  ?     Hundredths  ? 

3.  What  is  the  first  place  at  the  right  of  the  decimal 
point  ?     What    is    the    first   period    called  ?     The   second 
place  ?     The  second  period  ? 


40  SENIOR   ARITHMETIC. 

4.  What  is  a  mixed  decimal  ? 

5.  What  must  the  denominator  of  a  decimal  be  ? 

6.  What  is  the  effect  of  removing  the  decimal  point 
one  place  to  the  right  ?     To  the  left  ?     Two  places  to  the 
right  ?     Three  places  to  the  left  ? 

7.  What  is  the  effect  of  annexing  a  cipher  to  an  inte- 
ger ?     To  a  decimal  ?    Of  prefixing  a  cipher  to  an  integer  ? 
To  a  decimal  ? 

8.  How  do  we  reduce  decimals  to  common  fractions  ? 
Common  fractions  to  decimals  ? 

9.  Give  rules  for  adding,  subtracting,  multiplying,  and 
dividing  decimals. 

10.  How  do  we  locate  the  decimal  point  in  the  sum  ? 
In  the  remainder  ?     In  the  product  ?     In  the   quotient  ? 

11.  What  are  coins  ? 

12.  What  are  the  gold,  silver,  bronze,  and  nickel  coins 
used  in  the  U.  S.  ? 

13.  What  are  the  aliquot  parts  of  a  number  ?    What  are 
the  aliquot  parts  of  $1  ?     Of  100  ?     Of  1,000  ? 

14.  What  is   a  bill  ?      An  account  ?     A  creditor  ?     A 
debtor  ?     Tell  how  to  receipt  a  bill. 

119.    1.    Define  unit,  number,  the  unit  of  a  number,  ab- 
stract number,  concrete  number,  like  numbers. 

2.  Define  notation,  numeration,  Arabic  notation. 

3.  What  is  the  value  of  the  unit  figure  of  a  number  ? 
The  tens  ?     The  hundreds  ? 

4.  What  is  the  largest  number  which  can  be  expressed 
by  four  figures  ? 

5.  What  is  the  simple  value  of  a  figure  ?     The  local 
value  ? 


QUESTIONS. 

6.  What  name  is  given  to  the  first  period  at  the  right 
of  the  decimal  point  ?     The  second  ?     The  third  ? 

7.  What  is  addition?     What  kind  of  numbers  can  be 
added  ? 

8.  Define  subtraction,  minuend,  subtrahend,  remainder. 
What  is  a  proof  of  subtraction  ?     What  is  the  sign  of  sub- 
traction, and  where  placed  ? 

9.  What  is  a  parenthesis  ?     A  vinculum  ?     For  what 
are  they  used  ? 

10.  What  is  multiplication ?    The  multiplier?    The  mul- 
tiplicand ?     The  product  ? 

11.  The  multiplier  and  the  multiplicand  are  what  of  the 
product  ? 

12.  What  is  the  sign  of  multiplication  and  how  is  it 
used  ?      Define   division,   divisor,    dividend,    quotient,    re- 
mainder. 

13.  What  is  the  sign  of  division,  and  how  is  it  used  ? 

14.  Express  the  division  of  12  by  8  in  as  many  ways  as 
you  can. 

15.  To  what  terms  in  multiplication  do  the  divisor,  quo- 
tient, and  dividend  correspond  ? 

16.  How  do  you  find  the  dividend  when  the  divisor,  quo- 
tient, and  remainder  are  given  ? 

17.  When  is  the  quotient  an  abstract  number  ? 

18.  When  the  quotient  and  dividend  are  like  numbers, 
what  kind  of  a  number  is  the  divisor? 

19.  How  can  we  divide  when  the  divisor  is  10  ?    100  ? 
1000?     When  the  divisor  is  20?    50?    300? 

20.  Multiplying  both  dividend  and  divisor  by  the  same 
number  affects  the  quotient  how  ? 


42  SENIOR    ARITHMETIC. 

21.  Dividing    both    divisor   and    dividend  by   the   same 
number  affects  the  quotient  how  ? 

22.  Multiplying  the  dividend  affects  the  quotient  how  ? 
The  divisor  ?     Dividing  the  dividend  ?     The  divisor  ? 

23.  Define  exact  divisor,  factor,  prime  factor,  factoring. 

24.  How  can  you  find  the  prime  factors  of  a  number? 

25.  Define  divisor.     Common  divisor.    The  greatest  com- 
mon divisor.     Give  rule  to  find  greatest  common  divisor. 

26.  Define    multiple,    common    multiple,    least    common 
multiple.    Give  rule  for  finding  the  least  common  multiple. 

120,  Define  fraction,  fractional  unit,  unit  of  a  fraction, 
denominator,  numerator,  terms  of  a  fraction,  common  frac- 
tion, integer,  proper  fraction,  improper  fraction,  mixed  num- 
ber, simple  fraction,  compound  fraction,  complex  fraction. 

What  is  the  value  of  a  fraction  ? 

State  the  principles  of  fractions. 

What  is  it  to  reduce  a  fraction  ? 

How  are  fractions  reduced  to  lowest  terms  ?  To  highest 
terms  ? 

How  can  an  improper  fraction  be  reduced  to  a  whole 
or  a  mixed  number  ?  A  whole  or  a  mixed  number  to  an 
improper  fraction  ? 

What  are  like  fractions  ?     Unlike  fractions  ? 

How  can  fractions  be  reduced  to  others  having  a  common 
denominator  ?  A  least  common  denominator  ? 

How  can  two  or  more  fractions  be  added  ? 

How  can  the  sum  of  fractions  be  found  ?  Mixed  num- 
bers ? 

How  can  the  difference  of  fractions  be  found  ?  Mixed 
numbers  ? 

How  can  a  fraction  be  multiplied  by  a  fraction  ?  A  frac- 
tion by  an  integer  ? 


COMPOUND    NUMBERS.  43 

How  can  an  integer  be  multiplied  by  a  fraction  ?  By  a 
mixed  number  ? 

How  can  a  fraction  be  divided  by  a  fraction  ? 

How  do  you  reduce  a  complex  fraction  to  a  simple  frac- 
tion ? 

COMPOUND  NUMBERS. 

121.  A  number  composed  of  only  one  kind  of  unit  is 
a  Simple  Number ;  as,  5  pk.,  4  apples,  6. 

122.  A  Denomination  is  a  name  given  to  a  unit  of  measure 
or  of  weight. 

123.  A  number  composed  of  different  kinds  of  units  is 
a  Compound  Number ;  as,  3  bu.  2  pk.  1  qt. 

A  number  having  one  or  more  denominations  is  also  called 
a  Denominate  Number. 

124.  Reduction  is  the  process  of  changing  a  number  from 
one  denomination  to  another  without  changing  its  value. 

125.  Changing  to  a  lower  denomination  is  called  Reduction 
Descending ;  as,  2  bu.  3  pk.  =  88  qt. 

126.  Changing  to  a  higher  denomination  is  called  Reduc- 
tion Ascending ;  88  qt.  =  2  bu.  3  pk. 

127.  Linear  Measure  is  used   in  measuring  lines  or  dis- 
tances. 

TABLE. 

12    inches  (in.)  =  1  foot,         ft. 

3    feet  =  1  yard,        yd. 

5^  yards,  or  16 J-  feet  =  1  rod,  rd. 

40    rods  =  1  furlong,    fur 

320    rods,  or  5280  feet  =  1  mile,         mi. 
1  mi.  =  320  rd.  =  1760  yd.  =  5280  ft.  =  63360  in. 


44  SENIOR    ARITHMETIC. 

128.  Surveyors'  Measure  is  used  in  measuring  land. 

TABLE. 

7.92  inches  ==  1  link,     li. 
100  links     =  1  chain,  ch. 
80  chains  =  1  mile,     mi. 

NOTE.  —  A  surveyors'  chain  is  4  rods  long,  and  contains  100  links. 
A  chain,  or  steel  measuring  tape,  100  feet  long,  is  sometimes  used 
by  engineers. 

129.  Square  Measure  is  used  in  measuring  surfaces. 

TABLE. 

144  square  inches  =  1  square  foot,  sq.  ft. 

9  square  feet  =  1  square  yard,  sq.  yd. 

301  square  yards  j  = 

272i  square  feet     > 

160  square  rods  =  1  acre,  A. 

640  acres  =  1  square  mile,  sq.  mi. 

1  sq.  mi.  =  640  A.  =  102400  sq.  rd.  =  3097600  sq.  yd. 

130.  A  square  mile  of  land  is  called  a  Section. 

A  square  rod  is  sometimes  called  a  perch  (P.).  A  rood 
(K.)  is  40  sq.  rods. 

NOTE.  —  1  acre  =  43560  sq.  ft.  There  are  10  square  chains^  in 
an  acre. 

Roofing,  paving,  etc.,  are  often  estimated  by  the  Square, 
which  is  100  square  feet. 

131.  Cubic  Measure  is   used    in   measuring   volumes    or 
solids. 

TABLE. 

1728  cubic  inches  =  1  cubic  foot,  cu.  ft. 

27  cubic  feet  =  1  cubic  yard,  cu.  yd. 

16  cubic  feet  =  1  cord  foot,  cd.  ft. 

8  cord  feet,  or  128  cu.  ft.  =  1  cord,  C. 
1  cu.  yd.  =  27  cu.  ft.  =  46656  cu.  in. 


COMPOUND   NUMBERS.  45 

132.  Liquid  Measure  is  used  in  measuring  liquids. 

TABLE. 

4  gills  (gi.)  =  1  pint,       pt. 

2  pints          =  1  quart,     qt. 

4  quarts       =  1  gallon,    gal. 

1  gal.  =  4  qt.  =  8  pt.  =  32  gi. 

A  gallon  contains  231  cubic  inches. 

The  standard  barrel  is  31  \  gal.,  and  the  hogshead  63  gal 

129.    Apothecaries'  Fluid  Measure  is  used  in  mixing  medi- 
cines in  liquid  form. 

TABLE. 

60  minims  (TO)  =  1  fluid  dram,     f.  3. 
8  fluid  drams   =  1  fluid  ounce,    f .  §. 
16  fluid  ounces  =  1  pint  (0). 

133.  Dry  Measure  is  used  in  measuring  roots  grain,  vege- 
tables,  etc.  IABI^ 

2  pints     =  1  quart,       qt. 

8  quarts  =  1  peck,        pk. 

4  pecks    =  1  bushel,     bu. 

1  bu.  =  4  pk.  =  32  qt.  =  64  pt. 

The  bushel  contains  2150.42  cubic  inches. 

134.  Avoirdupois  Weight  is  used  in  weighing  all  common 
articles ;  as,  coal,  groceries,  hay,  etc. 

TABLE. 

16  ounces  =  1  pound,  Ib. 

(  1  hundred- weight,    cwt. 
100  pounds  =  •< 

(         or  cental,  ctl. 

20  cwt.,  or  2000  Ib.  =  1  ton,  T. 

1  T.  =  20  cwt.  =  2000  Ib.  =  32000  oz. 

The  Long  Ton  of  2240  pounds  is  used  at  the  U.  S.  Cus- 
tom-House and  in  weighing  coal  at  the  mines. 


46  SENIOR   ARITHMETIC. 

The  ounce  is  considered  as  16  drams. 

The  Avoirdupois  pound  contains  7000  grains. 

A  hundred-weight  is  sometimes  called  a  Cental. 

135.  Troy  Weight  is  used  in  weighing  gold,  silver,  and 
jewels.  TABLE. 

24  grains  (gr.)       =  1  pennyweight,    pwt. 
20  pennyweights  =  1  ounce,  oz. 

12  ounces  =  1  pound,  Ib. 

1  Ib.  =  12  oz.  =  240  pwt.  =  5760  grains. 

136.  Apothecaries'  Weight  is  used  by  druggists  and  physi- 
cians in  weighing  medicines  that  are  not  liquid. 

TABLE. 

20  grains  (gr.)  =  1  scruple,    sc.  or  9. 

3  scruples        =  1  dram,       dr.  or  3. 

8  drams  =  1  ounce,      oz.  or  %. 

12  ounces  =  1  pound,      Ib.  or  R>. 

1  Ib.  =  12  oz.  =  96  dr.  =  288  sc.  =  5760  gr. 

Dry  medicines  are  bought  and  sold  in  large  quantities 
by  avoirdupois  weight. 

Comparison  of  Weights. 

1  Ib.  Avoirdupois  =  7000  gr. 

1  oz.  Avoirdupois  =  437^  gr. 

1  Ib.  Troy  or  Apothecary     =  5760  gr. 
1  oz.  Troy  or  Apothecary  =  480     gr. 

137.  Measure  of  Time. 

TABLE. 


60  seconds  (sec.) 

=  1  minute,    min. 

60  minutes 

=  1  hour,        hr. 

24  hours 

=  1  day,          da. 

7  days 

=  1  week,       wk. 

365  days 

=  1  year,        yr. 

366  days 

=  1  leap  year. 

100  years 

=  1  century. 

COMPOUND   NUMBERS.  47 

The  Civil  Day  begins  and  ends  at  midnight. 

The  exact  time  in  which  the  earth  makes  one  revolution 
of  the  sun  is  365  da.  5  hr.  48  min.  49.7  sec.,  or  365£  days, 
nearly.  For  convenience  the  common  year  is  regarded  as 
365  days;  the  fraction  being  disregarded  until  it  amounts 
to  a  full  day,  which  is  in  four  years,  nearly.  Accordingly 
every  fourth  year  contains  366  days.  This  day  is  added 
to  the  shortest  month,  February,  and  the  year  in  which 
it  is  added  is  called  Leap  Year. 

But  365i  days  is  a  little  more  than  the  exact  year,  and 
we  have  added  a  little  too  much,  when  we  have  added  1 
day  to  every  fourth  year,  therefore  only  every  fourth  cen- 
tennial year  is  considered  as  leap  year.  This  nearly  cor- 
rects the  excess,  so  that  the  error  is  less  than  1  day  in 
about  3600  years. 

Every  year  divisible  by  4,  and  every  centennial  year 
divisible  by  400,  is  a  Leap  Year. 

Circular  or  Angular  Measure. 

138.  A  Circle  is  a  plane  figure  bounded  by  a  curved  line, 
every  point  of  which  is  equally  distant  from  the  centre. 

139.  The  bounding  line  of  a 
circle  is  the  Circumference. 

140.  Any  part  of  a  circum- 
ference is  an  Arc. 

A  to  C  and  B  to  D  are  arcs 
of  a  circle. 

141.  A  straight  line  through 
the   centre    of   a   circle   termi- 
nating at  the  circumference  is  A  circle. 
the  Diameter. 

142.  A  straight  line  from  the  centre  to  the  circumference 
is  the  radius ;  as,  E  to  'D,  or  E  to  C. 


48  SENIOJi   ARITHMETIC. 

143.  The  circumference  of  every  circle  is  divided  into 
360  equal  parts  called  Degrees,  each,  degree  into  60  parts 
called  Minutes,  and  each  minute  into  60  parts  called  Seconds. 

144.  An  Angle  is  the  difference  in  direction  between  two 
straight  lines.     The  point  of  meeting  is  the  Vertex.     The 
vertex  is  at  the  centre  of  a  circle,  and  the  angle  is  measured 
in  degrees  by  the  arc  between  its  sides.     Thus  B  D  is  the 
measure  of  the  angle  BED. 

TABLE   OF  CIRCULAR  MEASURE. 

60  seconds  (")  =  1  minute,  ' 

60  minutes         =  1  degree, 
360  degrees          =  1  circumference,    Cir. 

NOTE.— An  arc  of  90  degrees  or  £  of  a  circumference  is  called 
a  quadrant.  A  degree  upon  a  great  circle  of  the  earth  is  69.16 
statute  miles,  or  60  geographical  miles.  A  sign  is  an  arc  of  30 
degrees. 

146.  Federal  Money  is  the  currency  of  the  United  States. 

TABLE. 

10  mills  =  1  cent,    ct.        10  dimes    =  1  dollar,  $ 
10  cents  =  1  dime,   d.        10  dollars  =  1  eagle,    E. 

The  gold  coins  of  the  United  States  are  the  double-eagle, 
eagle,  half-eagle,  quarter-eagle,  and  one-dollar  piece. 

The  silver  coins  are  the  dollar,  half-dollar,  quarter-dollar, 
and  the  ten-cent  piece. 

The  five-cent  piece  is  nickel,  and  the  one-cent  piece  bronze. 

147.  English  or  Sterling  Money. 

TABLE, 

4  farthings  =  1  penny,  d. 
12  pence  =  1  shilling,  s. 
20  shillings  =  1  pound,  £,  or  1  sovereign. 

The  coin  which  represents  the  Pound  Sterling  is  the 
Sovereign,  equal  in  value  to  $4.8665. 


COMPOUND    NUMBERS.  49 

148.  Counting. 

TABLE. 

12  things  =  1  dozen,  doz. 

12  dozen   =  1  gross,  gr. 

12  gross    =  1  great  gross,    G.  gr. 

149.  Paper. 

TABLE. 

24  sheets  =  1  quire.  2  reams     =  1  bundle. 

20  quires  =  1  ream.  5  bundles  =  1  bale. 

REDUCTION   DESCENDING. 

150.  1.    Reduce  5  Ib.  6  oz.  12  pwt.  6  gr.  to  grains. 

5  Ib.  6  oz.  12  pwt.  6  gr. 

12 

Since  there  are  12  oz.  in  1  Ib.,  in  5  Ib.  there  are 

5  times  12  oz.  =  60  oz.  (add  6  oz.)  =  66  oz. 

_T  Since  there  are  20  pwt.  in  1  oz.,  in  66  oz.  there 

66  oz.  are  66  times  20  pwt.  =  1320  pwt.  (add  12  pwt.)  = 

20  1332  pwt. 

1320  Since  there  are  24  gr.  in  1  pwt.,  in  1332  pwt. 

12  there  are  1332  times  24  gr.  =  31968  gr.  (add  6  gr.) 

1332  pwt.  "  319?4  &- 

24  Reduce  to  lower  denominations  : 

2QQ±  2-    If  3^  2  ft.  9  in.  to  inches. 

31968  3-    ^G  rd.  4  yd.  2  ft.  to  feet. 

6  4.    3  mi.  75  rd.  4  ft.  to  inches. 


31974  gr.  5<    16  A   140  sq    rd   26  Sq.  yd.  to  square 

yards. 

6.  4  A.  15  sq.  rd.  4  sq.  ft.  to  square  inches. 

7.  50  ch.  45  li.  to  links. 

8.  16  cu.  yd.  25  cu.  ft.  900  cu.  in.  to  cubic  inches. 

9.  8  cd.  12  cu.  ft.  to  cubic  feet. 
10.    15  gal.  3  qt.  1  pt.  to  pints. 


SENIOR    ARITHMETIC. 

11.  4  0.  6  f.  I  3  f.  3  25  u  to  minims. 

12.  7  bu.  3  pk.  5  qt.  1  pt.  to  pints. 

13.  16^  bu.  to  quarts. 

14.  25  Ib.  5  oz.  16  pwt.  10  gr.  to  grains. 

15.  2  T.  6  cwt.  10  Ib.  14  oz.  to  ounces. 

16.  16  Ib.  5  %  4  3  2  9  11  gr.  to  grains. 

17.  28°  14'  18"  to  seconds. 

18.  £18  155.  Sd.  3  far.  to  farthings. 

19.  27  da.  18  h.  49  min.  to  seconds* 
3  wk.  48  min.  52  sec.  to  seconds. 

21.  How  many  quires  in  a  bundle  of  paper  ? 

22.  How  many  pints  in  a  cask  of  molasses  holding  84 
gallons  ? 

23.  How  many  articles  in  7  G.  gr.  5  gr.  ? 

24.  How  many  hours  in  10  years,  allowing  for  two  leap 
years  ? 

^S&.    How  many  inches  in  4^  rods  ? 

26.  What  is  the  cost  of  10  miles  of  telephone  wire  at  28 
cents  a  pound,  if  a  pound  measures  75  ft.  ? 

27.  Find  the  number  of  square  inches  in  a  square  yard ; 
square   feet  in   a   square   chain ;    cubic   inches   in   a  cubic 
yard. 

28y  How  many  hours  in  the  month  of  February,  1896  ? 
^/^b.    How  many  cubic  inches  in  5  gallons. 

30.  How  many  square  yards  in  4  sq.  miles  ? 

31.  How  many  square  feet  in  1\  acres  ? 

32.  How  many  ounces  in  3  Ib.  of  silver  ?     3  Ib.  of  iron  ? 

33.  If  I  buy  3  bu.  of  nuts  at  $4  a  bushel,  and  sell  them 
at  5/  a  pint,  how  much  shall  I  gain  ? 

34.  How  many  ounces  in  a  long  ton  ? 


COMPOUND    KUMBKKS.  51 

35.    At  $12  a  ton,  what  will  f  of  a  ton  of  hay  cost  ? 
In  1800  years  how  many  centuries  ? 

37.  If  you  can  count  sixty  a  minute,  how  long  will  it 
take  to  count  180000. 

38.  Through  how  many  degrees  does  the  hour-hand  of  a 
clock  pass  in  6  hours  ? 

39.  Through  how  many  degrees   does  the  minute-hand 
pass  in  6  hours  ? 

40.  What  will  3  reams  of  paper  cost  at  40/  a  quire  ? 

41.  Eeduce  3  mi.  4  fur.  20  rd.  5  yd.  2  ft.  8  in.  to  inches  ? 

42.  Eeduce  6  mi.  240  rd.  to  feet. 

43.  Reduce  3  A.  8  sq.  rd.  5  sq.  yd.  3  sq.  ft.  to  sq.  inches. 

44.  Reduce  16  cu.  yd.  9  cu.  ft.  3  cu.  in.  to  cu.  inches. 

45.  Reduce  58  cd.  to  cu.  feet. 

46.  Reduce  2  T.  3  ctl.  16  Ib.  to  ounces. 

47.  Reduce  3  Ib.  9  oz.  15  pwt.  12  gr.  to  grains. 
\^&T  Reduce  60  gal.  3  qt.  3  gi.  to  gills. 

49.  How  maiiy  sheets  in  5  bales  of  paper  ? 

50.  Reduce  3  wk.  6  da.  5  hr.  to  minutes. 

REDUCTION   ASCENDING. 

151.    1.    Reduce  1306  gills  to  higher  denominations. 

4  j  1306  gi.  Since  in  1  pt.  there  are  4  gi.,  in  1306  gi. 

2  /    3^6  pt    -4-  2  gi         there  are  as  many  pints  as  4  gi.  is  con- 

A  I — TAQ —  tained  times  in  1306  gi.,  or  326  pt.  and  2  gi. 
4  /     JLuo  Qt.  .     _ 

* * —  remainder. 

0  gal.  +  3  qt.  Since  in  j  qt>  there  are  2  pt.,  in  326  pt. 

40  gal.  3  qt.  2  gi.  there  are  as  many  quarts  as  2  pt.  is  con- 

tained times  in  326  pt.,  or  163  qt. 

Since  in  1  gal.  there  are  4  qt.,  in  163  qt.  there  are  as  many  gal- 
lons as  4  qt.  is  contained  times  in  163  qt.,  or  40  gal.,  and  3  qt. 
remainder. 

Therefore,  in  1306  gills  there  are  40  gal.  3  qt.  0  pt.  2  gi. 


52  SENIOR    ARITHMETIC. 

2.  How  many  rods  in  334  yd.  ? 

5J  yd.  334  yd.  Since  in  1  rd.  there  are 

22  5£   yd.,  in   334  yd.   there 

11  half  yd./ 668  half  yd.  are  as  many  rods  as  5*  ?d- 

— ™ — 5 o  T.    in — 7         is  contained  times  in  334 

60  rd.  + 8  half  yd.       yd.,  or  60  rd.,  and  4  yd. 

remainder.     334  yd.  -^  5i  yd.  =  668  half  yd.  -f-  11  half  yd.     8  half 
yd.  =  4  yd.     334  yd.  =  60  rd.  4  yd. 

3.  Reduce  225932  in.  to  miles,  etc. 

4.  How  many  miles  and  rods  are  there  in  35640  ft.  ? 

5.  Reduce  19922544  sq.  in.  to  higher  denominations. 

6.  Reduce  762051  cu.  in.  to  cu.  yards,  etc. 

7.  How  many  cords  in  7424  cu.  ft. 

8.  Reduce  69056  oz.  to  tons,  etc. 

9.  Reduce  21076  gr.  to  higher  denominations. 

10.  Reduce  1947  gi.  to  gallons,  etc. 

11.  How  many  bales  in  24000  sheets  of  paper  ? 
^}ti£  Reduce  39180  min.  to  weeks,  etc. 

13.  Reduce  5762  far.  to  higher  denominations.   • 

14.  Reduce  84623''  to  higher  denominations. 

15.  Reduce  62341  M.  to  higher  denominations. 

16.  How  many  chains,  etc.,  in  13025  li.  ? 

17.  How  many  bushels,  etc.,  in  35842  pints  ? 

18.  How  many  pounds,  etc.,  (Troy)  in  32563  gr.  ? 

19.  Reduce  39632  gr.  to  lb.,  etc.  (Apoth.). 

20.  How  many  tons,  etc.,  in  35682  lb.  ? 

21.  A  box  contains  75832  pens.     How  many  Gr.  gross, 
etc.,  in  the  box  ? 

22.  Change  1384  dry  pints  to  higher  denominations. 

23.  In  139843  sq.  in.  how  many  miles,  rods,  etc.  ? 

24.  How  many  cords  of  wood  in  3692  cu.  feet  ? 


COMPOUND    NUMBERS.  53 

REVIEW    PROBLEMS. 

152.    1.    Bought  2  gal.  8  oz.  of  fluid  extract  at  20/  an 
ounce,  and  sold  it  at  15/  an  ounce.     What  was  lost  ? 

2.  How  many  minims  are  there  in  10  fluid  ounces  (f.  5), 
7  fluid  drachms  (f.  3)  ? 

3.  Find  the  difference  in  value  between  4  gal.  of  am- 
monia water  at  10  cents  a  pint  and  8  ounces  of  cinnamon 
water  at  5  cents  an  ounce. 

4.  The  pendulum  of  a  certain  clock  beats  seconds.    How 
many  times  will  it  tick  in  1  day,  9  hours,  25  minutes  ? 

5.  How  many  degrees  in  3492.58  statute  miles,  measured 
on  the  equator,  a  degree  being  equal  to  69.16  statute  miles  ? 

6.  How  many  degrees  of  longitude  will  a  steamship  pass 
through,  sailing  due  west  on  the  equator,  at  the  rate  of  15 
knots  an  hour  for  5  days  ? 

NOTE.  —  A  knot  =  1  geographic  mile  or  minute. 

7.  Find  cost  of  each  of  following : 

(a)    5  gallons,  3  qt.  1  pt.  of  molasses  at  20/  a  gallon ; 
(I)    10  acres,  50  sq.  rd.  of  land  at  $50  an  A. 

8.  What  will  it  cost  to  build  112  rd.  3  yd.  of  fence  at 
48/  a  yard  ? 

9.  If  a  man  steps  2^  ft.  at  each  step,  how  many  miles 
will  he  travel  in  stepping  4820  times  ? 

10.  If  17  ft.  is  |  of  the  height  of  a  tree,  how  high  is  the 
tree? 

11.  Reduce  6f  mi.  317  rd.  4  yd.  2  ft.  to  feet. 

12.  Change  16571  ft.  to  miles. 

13.  At  $3.20  a  bu.   how  many  quarts  cvf  nuts  can  be 
bought  for  $4.80  ? 

14.  How  many  pint  bottles  of  camphor  may  be  filled 
from  96  fluid  ounces  (f.  3)  ? 


54  SENIOR   ARITHMETIC. 

15.  Find  the  cost  of  the  following :  4  oz.  iodine  at  10/, 
8  oz.  spts.  camphor  at  5/,  10  oz.  aqua  ammonia  at  10^,  14 
oz.  cinnamon  water  at  5/. 

16.  Reduce  4  bu.  3  pk.  7qt.  1  pt.  to  pints. 

17.  How  many  quart  boxes  will  hold  2  bu.  3  pk.  5  qt.  of 
berries  ? 

18.  If  4  bu.  of  berries  are  bought  for  $.70  per  bushel 
and  sold  for  $.05  per  quart,  what  is  the  gain  or  loss  ? 

19.  Both  sides  of  a  railroad  track  are  fenced  with  wire  for 
40  yards.     What  is  the  cost  of  the  fence  at  4/  a  foot  ? 

v2tT   What  will  8  Ib.  6  oz.  of  sugar  cost  at  8/  a  pound  ? 

21.  When  pens  are  bought  at  75/  a  gross,  and  sold  at  2 
for  3/,  what  is  the  gain  ? 

22.  If  a  man  can  walk  10  miles  in  2  hours,  how  far  can 
he  walk  in  6  hours  ?  in  30  minutes  ?  in  50  minutes  ? 

23.  What  will  \  bu.  berries  bring  at  8/  a  quart  ? 

24.  A   silver   chain  weighs  18  pwt. ;  what  is  its  value, 
when  silver  is  worth  $.65  an  ounce  ? 

25.  What  will  24  qt.  of  milk  cost  at  20/  a  gallon  ? 

26.  If  I  buy  peanuts  at  5/  a  quart,  and  retail  them  so 
as  to  gain  $6.40  on  4  bushels,  what  do  I  sell  them  for  ? 

27.  At  4  pens  for  3  cents  what  will  1  great  gross  cost  ? 

28.  How  many  table-forks,  each  weighing  2^  oz.,  can  be 
made  from  4  Ib.  4  oz.  10  pwt.  of  silver. 

29.  In  |  of  a  gallon  how  many  pints. 

30.  How  many  rods  of  fence  will  enclose  a  mile  square 
of  land  ? 

31.  What  is. the  cost  of  1  yd.  and  27  inches  of  fringe  at 
60  cents  a  yard. 

32.  How  many  rods  of  fence  are  required  to  enclose  a 
lot  that  is  20  rods  wide  and  three  times  as  long  ? 


COMPOUND   NUMBERS.  55 

33.  Kequired  the  distance  around  a  room  that  is  13  feet 
long  and  15  feet  wide. 

34.  A  shoe-box  is  4  in.  deep,  6  in.  wide,  and  12  in.  long. 
How  much  twine  will  it  take  to  wind  twice  around  the  box 
each  way  to  hold  on  the  cover,  allowing  6  inches  for  tying. 

35.  I  have  a  lawn  that  is  30  ft.  by  70  ft.,  and  wish  to  lay 
a  board  walk  around  it  that  is  3  ft.  6  in.  in  width.     What 
is  the  distance  around  the  walk,  outside  measurement  ? 

153.  A  Denominate  Fraction  is  a  fraction  having  a  de- 
nomination. 

154.  To  reduce  Denominate  Fractions  to  Integers  of  Lower 
Denominations. 

1.  Reduce  f  of  a  mile  to  rods,  yards,  feet,  etc. 

SOLUTION.  —  f  of  320  rd.  =  JL«7<L°  rd.  =  228f  rd. 
f  of  V  yd.  =  ft  yd.  =  3}  yd. 
\  of  3  ft.  =  Q\  ft. 
f  of  12  in.  =  -376  in.  =  5J  in. 
f  of  a  mile  =  228  rd.,  3  yd.,  0  ft.,  5£  in. 

NotE.  —  The  same  process  applies  to  denominate  decimals. 

2.  Reduce  .87  bu.  to  pecks,  quarts,  etc. 

.87  of  4  pk.  =  3.48  pk.  87  bu. 

4 
.48  of  8  qt.  =  3.84  qt.  3.48 

8 
.84  of  2  pt.  =  1.68  pt.  3.84 

2 
.87  bu.  =  3  pk.,  3  qt.,  1.68  pt.  1.68 

Rule. —  Change  the  given  fraction  (or  decimal')  to  the  next 
lower  denomination.  Treat  the  fractional  (or  decimal ) 
part  of  the  product  in  the  same  way,  and  so  proceed  to 
the  required  denomination. 


56  SENIOR   ARITHMETIC. 

Keduce  to  integers  of  lower  denominations. 

3.  §  of  a  mile.  9.  .375  of  a  month. 

4.  |  of  an  acre.  10.  .3125  of  a  gallon. 

5.  f  of  a  pound  (Troy).  11.  .4267  of  an  acre. 

6.  f  of  a  ton.  12.  .2364  of  a  ton. 

7.  |  of  a  gallon.  13.  .363  of  a  sign. 

8.  |  of  a  mile.  14.  .51625  of  a  mile. 

15.  Reduce  J|  mi.  to  lower  denominations. 

16.  Change  f  of  a  year  to  months  and  days. 

17.  In  fo  gal.  how  many  qt.  and  pt.  ? 

18.  Reduce  T225T  Ib.  to  oz.  and  dr. 

19.  T9¥  acres  are  equal  to  how  many  sq.  rods,  etc.  ? 

20.  Reduce  f  *  bu.  to  integers  of  lower  denominations. 

21.  What  is  the  value  of  f  of  f  of  a  hhd.  in  integers  of 
lower  denominations  ? 

22.  What  is  the  value  of  T75  of  an  acre  in  integers  of 
lower  denominations  ? 

23.  Reduce  £^  to  integers  of  lower  denominations. 

24.  What  is  the  value  of  ^  of  1^  of  a  mile  ? 

155.    To  reduce  denominate  numbers  to  Fractions  of  Higher 
Denominations. 

1.  Reduce  2  qt.  1  pt.  2  gi.  to  the  fraction  of  a  gallon. 
SOLUTION.  —     2  gi.  -f-  4  =  f  pt.  =  |  pt. 

18 rpt.  =  |  pt.  -r  2  =  |  qt. 

2|  qt.  =  1|  qt.  -=-  4  =  \16  gal.     Ans. 

2.  Reduce  2  qt.  1  pt.  2  gi.  to  the  decimal  of  a  gallon. 

Rule.  —  Change  the  number  of  the  lowest     4  /2  gi. 
denomination  to  a  fraction  (or  deci-     2  /1. 5  pt. 
mal )   of  the  next  higher,  write  this        4  /2.75  qt. 
fraction   (or  decimal)  as   a  part  of  .6875  gal. 

the  number  of  that  higher  denomination,  and  reduce 


COMPOUND    NUMBEliS.  57 

this    number   in   like  manner,   and  so  proceed   to   the 
req  uired  denom ination. 

•   3.    Reduce  213  rd.  1  yd.  2  ft.  6  in.  to  a  fraction  of  a 
mile. 

4.  What  fraction  of  an  acre  is  3  R.  13  sq.  rd.  10  sq.  yd. 
108  sq.  in.  ? 

5.  What  part  of  a  year  is  273  da.  18  hr.  ? 

6.  Reduce  to  a  fraction  of  a  pound  8  oz.  11  pwt.  lOf  gr. 

7.  What  part  of  a  ton  is  857  Ib.  2f  oz.  ? 

8.  Change  3  fur.  19  rd.  5  yd.  1  ft.  4.7328  in.  to  the  deci- 
mal of  a  mile. 

9.  Reduce   1  da.  14  hr.  24  min.   to  the   decimal  of  a 
month. 

10.  What  decimal  of  a  gallon  is  1  qt.  2  gi.  ?• 

11.  Reduce  68  sq.  rd.  8  sq.  yd.  2  sq.  ft.  7.488  sq.  in.  to 
the  decimal  of  an  acre. 

12.  What  decimal  of  -a  pound  Troy  is  6  oz.  3  pwt.  21.6 
gr.  ? 

13.  Reduce  131  da.  18  hr.  21  min.  36  sec.  to  the  decimal 
of  a  year. 

14.  Reduce  2  qt.  If  gi.  to  the  fraction  of  a  gallon. 
v#C    What  fraction  of  a  mile  is  71  rd.  1  ft.  10  in.  ? 

16.  Reduce  12  da.  34  min.  14}  sec.  to  the  fraction  of 
a  month. 

17.  What  decimal  of  a  ton  is  4  cwt.  72  Ib.  128  oz.  ? 

18.  Reduce  48  cu.  ft.  1636.7616  cu.  in.  to  the  decimal  of 
a  cord. 

19.  What  decimal  of  a  circle  is  10°  53'  24"  ? 

20.  Reduce  4  fur.  5  rd.  1  yd.  3.6  in.  to  the  decimal  of  a 
mile. 


58  SENIOR    ARITHMETIC. 

21.  Reduce  6  pwt.  to  a  fraction  of  a  pound. 

22.  3  qt.  1  pt.  2  gills  are  what  part  of  a  peck  ? 

23.  Change  6  rd.  4  yd.  1  ft.  to  the  fraction  of  a  mile. 

24.  What  part  of  a  cord  of  wood  are  8  cu.  ft.  ? 

25.  Reduce  5  gross  7  doz.  to  the  fraction  of  a  score. 

To  find  what  part  one  denominate  number  is  of  another. 

1.  What  part  of  2  gal.  1  qt.  1  pt.  is  3  qt.  1  pt.  1  gi.  ? 

3  qt.  1  pt.  1  gi.  =  29  gi. 
2  gal.  1  qt.  1  pt.  =  76  gi. 

The  question  now  is,  29  gi.  is  what  part  of '76  gi.  ? 
29  gi.  is  f- I  of  76  gi.     .4ns. 

NOTE.  —  To  find  the  decimal  part,  divide  numerator  by  denomi- 
nator. 

2.  What  part  of  5  Ib.  9  oz.  3  pwt.  is  2  Ib.  8  oz.  6  pwt. 
10  gr.  ? 

3.  What  part  of  3  mi.  24  rd.  5  yd.  is  2  mi.  34  rd.  4  yd.  ? 

4.  What  part  of  3  da.  5  hr.  22  min.   is  1  da.  10  hr. 
3  min.  12  sec.  ? 

5.  What  decimal  of  3  gal.  2  qt.   1  pt.  is  2  gal.  2   qt. 
2  pt.  ? 

6.  What  decimal  of  4  T.  5  cwt.  10  Ib.  is  2  T.  6  cwt.  13 
Ib.  ? 

7.  What  part  of  a  rod  is  4  yd.  2  ft.  7  in.  ? 

8.  What  part  of  6  rods  is  }  of  7  feet  ? 

9.  What  part  of  3|  mi.  is  160  rd.  5  yd.  ? 

10.  ^  pint  is  what  part  of  a  bushel  ? 

11.  What  decimal  of  8  bu.  3  pk.  4  qt.  is  4  bu.  1  pk. 
5  qt. 

ADDITION   OF   COMPOUND   NUMBERS. 

156.    1.    Add  14  Ib.  5  oz.  17  pwt.  12  gr.,  18  Ib.  10  oz. 
14  gr.,  6  Ib.  4  oz.  8  pwt.  16  gr. 


COMPOUND   NUMBERS.  59 

lb.      oz.    pwt.    gr.  SOLUTION.  —  The  sum   of  the  grains  =  42 

14        5      17      12       gr-  —  1  pwt.  18  gr.     We  place  the  18  gr.  under 


18 

10 

0 

14 

the  column 

of  grains, 

and  add  the 

1  pwt.  to 

6 

4 

8 

16 

the  column 

of  pennyweights, 

,     Add 

the  other 

39 

8 

6 

18 

columns  in 

like  manner. 

rd. 

yd. 

ft. 

rd. 

ft. 

in. 

2. 

17 

4 

1 

3.      6 

12 

6 

12 

4 

2 

4 

14 

11 

6 

5 

2i 

17 

15 

<) 

8 

3   - 

.  o 

6 

12 

8 

46 

H 

H 

36 

5J 

10 

H 

=  1  yd. 

6  = 

i   ft. 

46     2       0  36 


tons,  i 

L5\Vt. 

lb. 

oz. 

yr. 

da. 

hr.  min. 

sec. 

4. 

14 

13 

65 

15 

7.   18 

345 

13 

37 

15 

13 

17 

88 

11 

87 

169 

12 

16 

28 

46 

16 

86 

13 

316 

144 

20 

53 

18 

14 

15 

57 

6 

13 

360 

21 

57 

15 

11 

17 

85 

15 

bu. 

pk. 

qt.   pt. 

deg.  min. 

sec. 

8.  40 

2 

6 

1 

5. 

29 

59 

59 

89 

1 

3 

0 

15 

45 

42 

75 

2 

1 

1 

18 

11 

40 

69 

2 

3 

0 

13 

19 

17 

49 

1 

2 

1 

65 

3 

1 

1 

sq.  yd.  sq 

.  ft. 

sq.  in. 

6. 

45 

8 

113 

cd. 

cd.  ft.  cu. 

ft. 

45 

3 

112 

9.  5 

7 

0 

75 

8 

139 

2 

2 

12 

49 

0 

115 

0 

6 

15 

589 

8 

90 

n 

0 

0 

3 

0 

2 

60  SENIOR    ARITHMETIC. 

10.  Find  the  sum  of  3  T.  15  cwt.  25  Ib.  9  oz.,  4  T.  17 
cwt.  30  Ib.  10  oz.,  6  T.  18  cwt.  15  Ib.  12  oz.,  2  T.  12  cwt. 
20  Ib.  16  oz. 

11.  Find  the  sum  of  7  hr.  30  inin.  45  sec.,  12  hr.  25  min. 
30  sec.,  20  hr.  15  min.  33  sec.,  10  hr.  27  min.  46  sec. 

12.  Add  10  mi.  101  rd.  3  yd.  2  ft.  11  in.,  16  mi.  4  yd. 
6  in.,  3  mi.  560  rd.  3  ft.,  175  rd.  4  ft.  7  in. 

13.  Add  3  A.  50  sq.  rd.  25  sq.  yd.  10  sq.  ft.  102  sq.  in., 
5  A.  110  sq.  rd.  30  sq.  yd.  8  sq.  ft.  34  sq.  in.,  6  A.  75  sq. 
rd.  14  sq.  yd.,  7  sq.  ft.,  82  sq.  in.,  7  A.  215  sq.  rd.,  17  sq. 
yd.  16  sq.  ft.  53  sq.  in. 

14.  Find  the  sum  of  18  cd.  6  cd.  ft.  12  cu.  ft.,  19  cd.  4 
cd.  ft.  6  cu.  ft.,  24  cd.  2  cd.  ft.  1  cu.  ft. 

15.  Find  the  sum  of  18  T.  18  Ib.  12  oz.,  16  cwt.  21  Ib., 
14  cwt.  75  Ib.  10  oz. 

16.  AVhat  is  the  entire  length  of  a  railway  consisting  of 
5  different  lines  measuring  respectively  160  mi.  185  rd.  2 
yd.,  97  mi.  63  rd.  4  yd.,  126  mi.  272  rd.  3  yd.,  67  mi.  199 
rd.  5  yd.,  and  48  mi.  266  rd.  5  yd.  ? 

17.  A  merchant  sold  48  gal.  3  qt.  1  pt.  of  coal  oil  and 
had  15  gal.  1  qt.  1  pt.  left.     What  quantity  had  he  at  first  ? 

18.  A  starts  from  a  point  in  Lat.  21°  25'  35"  N.  and 
travels  north  24°  36'  45".    At  what  latitude  does  he  arrive  ? 

19.  Find  the   difference    in    longitude  between  a  poinl 
46°  15'  30"  E.  and  a  point  21°  18'  16"  AY. 

NOTE.  —  When  one  place  is  in  east  and  the  other  in  west  longi 
tude,  their  difference  in  longitude  is  the  sum  of  their  longitudes. 

20.  Charles  walks  5  mi.  15  rd.  2  ft.  north  of  the  school 
house,  and  James  6  mi.  28  rd.  5  yd.  south.      How  far  are 
they  apart  ? 

21.  Find  the  sum  of  f  mi.  35  rd.  4 3  rd. 


COMPOUND    NUMBERS.  61 

NOTE.  —  Reduce  each  to  integers  of  lower  denominations,  then 
add. 

22.  Add  |  bu.  17f  pk.  4f  pt.?  6  bu.  3f  pk.  2  qt.,  1  bu. 
-1  pk.  5  qt. 

23.  What  is  the  sum  of  f  T.  |  cwt.  and  f  Ib.  ? 

SUBTRACTION   OF   COMPOUND   NUMBERS. 

Ib.    oz.   pwt.   gr.  SOLUTION.  —  15    gr.  - 

157.    l.  From  6      2     14     15       12  Sr-  -  3  Sr-    As  we  can' 

Take    4    10     18     12       nofc  take  18  Pwt-  from  14 
~ — — —       pwt.,  we  take  1  oz.,  which 

equals  20  pwt.,  and  add  to 

the  14  pwt.  =  34  pwt. ;  34  pwt.  —  18  pwt.  =  16  pwt.  We  have  taken 
1  oz.  from  the  2  oz.,  leaving  1  oz. ;  but  as  we  cannot  take  10  oz.  from 
1  oz.,  we  take  1  Ib  =  12  oz.,  and  add  it  to  1  oz.  =  13  oz.,  from  which 
take  10  oz.  =  3  oz.  Since  we  took  1  of  the  6  Ib.,  we  have  5  left; 
from  which  take  4  Ib.  =  1  Ib. 

3. 

hr.  min.  sec. 

5      54  30 

1      17  50 

4.  5. 

gal.  qt.  pt.  gi.  A.  R.  sq.  rd.  sq.  yd.  sq.  ft. 

From  39  2  2  1  5  1  39    15     7 

Take  10  2  3  3  2 2  26    21    _8 

6.  7. 

da.   hr.  min.  sec.  T.  cwt.  Ib.  oz. 

200  17  54  36  20  15  75  10 

135  20  24  48  5  16  25  12 

7.  From  260  ini.  take  23  mi.  7  fur.  25  rd.  5  yd.  2  ft. 
10  in. 


2. 

A. 

sq.  rd. 

sq.  ft. 

From   10 

50 

7 

Take      4 

106 

5 

62 


SENIOR    ARITHMETIC. 


8.  A  man  having  ^  an  acre  of  ground,  sold  25  sq.  rd.  11 
sq.  yd.  8  sq.  ft.  to  one  man,  and  50  sq.  rd.  9  sq.  yd.  4  sq.  ft. 
to  another.     How  much  land  had  he  left  ? 

9.  From  12  cwt.  subtract  9  cwt.  14  lb.  12  oz. 

10.  From  a  hogshead  of  molasses,  25  gal.  3  qt.   2  pt. 
were  drawn  at  one  time,  and  at  another  time  10  gal.  1  pt. 
How  many  gallons  remained  ? 

11.  From  2  bu.  3  pk.,  1  bu.  2  pk.  6  qt,  were  sold.     How 
much  remained  ? 

12.  An  apothecary  bought  2  lb.  of  quinine,  and  sold  1  lb. 
3  oz.  5  dr.  2  sc.  11  gr.     How  much  had  he  left  ? 

13.  What  is  the  difference  in  longitude  between  New 
York  (74°  0'  3"  W.)  and  San  Francisco  (122°  25'  40"  W.). 

NOTE.  — When  both  places  are  in  east  or  in  west  longitude,  their 
difference  in  longitude  is  found  by  subtraction. 

14.  Charles  walks  25  mi.  4  rd.  2  yd.  south  of  the  school, 
and  Henry  16  mi.  160  rd.   3  yd.  in  the  same  direction. 
How  far  are  they  apart  ? 

15.  From  |  of  a  mile  take  16£  rods. 

NOTE.  —  Change  both  to  integers  of  lower  denominations,  then 
subtract. 

16.  Rome  is  in  longitude  12°  28'  40"  E.,  and  Paris  in 
long.  2°  20'  14"  E.      What  is  their  difference   in   longi- 
tude ? 

17.  From  f  of  a  pound  Avoir,  take  3|  oz. 

18.  From  16|  bu.  take  1\  pk. 

19.  From  .325  T.  take  6.54|  cwt. 

20.  From  .7  of  a  rod  take  4  yd.  2  ft.  8  in. 

21.  From  22  da.  16  hr.  20  min.  take  2i  weeks. 


COMPOUND   NUMBERS.  63 

DIFFERENCE   BETWEEN   DATES. 

158.    1.   Find  the  time  from  Jan.  25, 1842,  to  July  4, 1896. 

1896          7  It  is  customary  to  consider  30  days  to 

1842          1          25.  a  month.     July  4,   1896,  is  the   1896th 

54  yr.  5  mo.   9  da.        yr.  7th  mo.  4th  da.,  and  Jan.  25,  1842, 
is  the  1842d  yr.  1st  mo.  25th  da.     Sub- 
tract, taking  ;-JO  da.  for  a  month. 

2.  What  is  the  exact  number  of  days  between  Dec.  16, 
1895,  and  March  12,  1896  ? 

Dec.   15  Do  not  count  the  first  day  mentioned. 

Jan.  31  There  are  15  days  in  December,  after  the 

Feb.   29  16th.     January  has  31  days,  February  29 

Mar.  12  (leap  year),  and  12  days  in  March  ;  making 

87  days.  87  days.     Ans. 

3.  How  much  time  elapsed  from  the  landing  of  the  Pil- 
grims, Dec.  11,  1620,  to  the  Declaration  of  Independence, 
July  4,  1776  ? 

4.  How  much  time  elapsed  from  the  beginning  of  the 
Civil  War,  April  14,  1861,  to  the  close  of  the  war  April  9, 
1865? 

5.  Washington  was  born  Feb.  22,  1732,  and  died  Dec. 
14,  1799.     How  long  did  he  live  ? 

6.  Washington  was  first  inaugurated  April  30,   1789. 
How  long  ago  was  his  inauguration  ? 

7.  How  much  time  will  have  elapsed  since  Columbus 
discovered  America,  Oct.  12,  1492,  to  your  next  birthday  ? 

8.  Mr.  Smith  gave  a  note  dated  Feb.  25,  1896,  and  paid 
it  July  12,  1896.     Find  the  exact  number  of  days  between 
its  date  and  time  of  payment. 

9.  A   carpenter   earning    $2.50    per    day,    commenced 
Wednesday  morning,  April  1,  1896,  and  continued  work- 
ing every  week  day  until  June  (>.     How  much  did  he  earn  ? 


64  SENIOR   ARITHMETIC. 


Fred  was  born  Dec.  20,  1875 ;  how  old  is  he  now  ? 

11.  How  much  time  has  elapsed  since  George  Washing- 
ton was  15 1  years  old  ? 

12.  Gen.   Grant   was  born   April   27,   1822.      How  old 
would  he  be  if  he  were  alive  to-day  ? 

13.  How  long  since  Lee  surrendered  to  Gen.  Grant  ? 

14.  Find  the  exact  number  of  days  between  Jan.  10, 
1896,  and  May  5,  1896. 

15.  When  can  a  boy  who  was  born  May  5,  1882,  cele- 
brate his  25th  birthday  ? 

16.  John  goes  to  bed  at  9.15  P.M.  and  gets  up  at  7.10  A.M. 
How  many  minutes  does  he  spend  in  bed  ? 

MULTIPLICATION   OF  COMPOUND   NUMBERS 

159.    1.    Multiply  4  yd.  2  ft.  8  in.  by  8. 

yd.     ft.    in.  8  times  8  in  =  64  in  =  5  ft    4  in      Place  the  4 

428          in.  under  the  inches  column,  and  reserve  the  5  ft., 

8         to  be  added  to  the  product  of  2  ft.  by  8,  which 

39       0       4         equals  16  ft.  (add  5  ft.)  =  21  ft.    21  ft.  -j-  3  =  7  yd., 

with  no  remainder.     Add  7  yd.  to  the  product  of 

4  yd.  by  8  =  32  yd.  (add  7  yd.)  =  39  yd. 

2.  gal.  qt.   pt.  gi.  bu.   pk.  qt.  pt. 
31      3     2     3  12     3      2     1 

5  8 

3.  If  a  man  travel  at  the  rate  of  60  mi.  240  rd.  16  ft.  in 
one  day,  how  far  will  he  travel  in  ten  days  ? 

4.  A  man  owns  6  farms,  each  containing  75  A.  49  sq.  rd. 
25  sq.  yd.  of  land.     How  much  land  in  all  the  farms  ? 

5.  If  6  loads  of  hay  weigh  6  T.  18  cwt.  75  lb.,  how 
much  will  48  loads  weigh  ? 

NOTE.  —  48  loads  will  weigh  8  times  as  much  as  fi  loads. 


COMPOUND    NUMBERS.  65 

6.  If  12  spoons  weigh  3  Ib.  8  oz.  15  pwt.,  how  much 
will  one  gross  of  spoons  weigh  ? 

7.  How  much  oil  will  7  barrels  hold  if  each  barrel  con- 
tains 35  gal.  2  qt.  ? 

8.  What  is  the  value  at  $4  per  cord  of  10  piles  of  wood, 
each  containing  5  cd.  5  cd.  ft.  12  cu.  ft.  ? 

9.  What  is  the  weight  of  15  packages,  each  weighing 
1  Ib.  4  oz.  (Avoir.)  ? 

10.  If  a  bicyclist  travels  75  mi.  140  rd.  in  one  day,  how 
far  can  he  travel  in  ten  days  ? 

11.  In  a  watch-chain  there  are  2  oz.  12  pwt.  15  gr.  of 
gold.     How  much  gold  is  required  for  25  such  chains  ? 

12.  A  farmer  has  six  bins,  each  containing  60  bu.  2  pk. 
of  wheat.     How  much  wheat  has  he  ? 

13.  If  a  train  is  run  for  8  hours  at  the  average  rate  of 
50  mi.  30  rd.  10  ft.  per  hour,  how  great  is  the  distance 
covered  ? 

14.  It  takes  John  Smith  5  hr.  20  min.  11  sec.  to  plough 
one  acre  of  ground.     At  the  same  rate,  how  long  will  it 
take  him  to  plough  4  acres  ? 

15.  4  gal.  3  qt.  l.pt.  X  11  =  ? 

16.  2  A.  40  sq.  rd.  16  sq.  yd.  X  20  =  ? 

DIVISION    OF   COMPOUND   NUMBERS. 

160.    1.  Divide  16  Ib.  9  oz.  17  pwt.  8  gr.  by  10. 

Ib.  oz.  pwt.    gr.  SOLUTION.  -  ^   of   16  Ib.  =  1,  and 

10  /16     9      17       8  6  Ib.  remaining.     6  Ib.  =  72  oz.     72  oz. 

18         3     17^        +9  oz.  =  81  oz.     TV  of  81  oz.  =  8  oz., 

with    1    oz.    remaining,  =  20   pwt.,    to 

-which  add  17  pwt.,  =  37  pwt.  Tlff  of  37  pwt.  =  3  pwt.,  with  7  pwt. 
remaining,  ==  168  gr.,  to  which  add  8  gr. ;  and  taking  Jff  of  the  sum, 
we  have  17T6o  gr. 

When  the  divisor  is  large,  employ  long  division. 


66  SENIOR    ARITHMETIC. 

2.    Find  ^  of  42  rd.  4  yd.  2  ft.  8  in. 

rd.  yd.  ft.  in. 

35/42     4      2     8  ( 1  rd.  -J:>  of  42  rd.  =  1  rd. ;  remainder, 

35  7  rd.  =  38^  yd. ;  add  4  yd.  =  42i  yd. 

3^  of  42^  yd.  =  1  yd.;  remainder,  7i  yd.,  =  22^ 

5L  ft.;    add  2   f t.  =  24i  ft.     3V,   of  24i  ft.  -  0  ft. 

24^  ft.  =  294  in. ;  add  8  in.  =  302  in.     ^  of  302 

Og-  .  022    ' 

35 

38^  NOTE.  — When  both  dividend  and  divisor  are 

-|-  4  compound,  reduce  them  to  the  same  de- 

35/42^.yd.  (1  yd.          nomination,  and  divide.     The  quotient 
35"  will  be  abstract. 

3.  Divide  169  bu.  3  pk.  5  qt. 

by  7. 

4.  If  a  man  travelled  607  mi. 
169   rd.   11   ft.  6   in.  in  10   days, 
what  average  distance  did  he  travel 
in  1  day  ? 

5.  If  one  gross  of  spools  weighs  44 
Ib.  9  oz.,  how  much  will  one  dozen 
weigh  ? 

0  .  6.    If  one  bottle  holds  1  pt.  3  gi., 

i  rci.  i  VQ.  oj^  in.  * 

.  how  many  dozen  bottles   will  be   re- 

quired to  hold  65  gal.  2  qt.  1  pt.  ? 

7.  A  man  has  451  A.  138  sq.  rd.  29  sq.  yd.  of  land, 
which  he  wishes  to  divide  equally  among  his  six  children. 
How  much  land  will  each  child  receive  ? 

8.  If  12  persons  share  equally  in  the  contents  of  a  bin 
containing  20  bu.  2  pk.  4  qt.  of  apples,  what  is  the  share 
of  each  ? 

9.  If  the  entire  area  of  24  equal  fields  is  242  A.  20  sq. 
rd.  15  sq.  yd.,  what  is  the  size  of  each  field  ? 


COMPOUND   NUMBERS.  67 

10.  A  man  walked  50  mi.   71   rd.  2  yd.   in  15   hours. 
What  was  his  rate  per  hour  ? 

11.  If  it  takes  a  man  12  hr.  35  min.  15  sec.  to  walk  45 
miles,   what  is   the   average    time   taken  for  each   mile  ? 
(Divide  by  the  factors  of  45.) 

12.  When   $12   will   buy   11  gal.  2  qt.    1  pt.  of  maple 
sirup,  how  much  will  $1  buy  ? 

13.  A  man  travelled  100  miles  in  9  hours.     What  was 
the  average  rate  per  hour  ? 

14.  If  a  horse  eats  12  qt.  of  oats  per  day,  how  long  will 
10  bu.  1  pk.  4  qt.  last  him  ? 

15.  If  a  package  weighs  4  cwt.  15  lb.,  how  many  such 
packages  will  it  take  to  weigh  3  T.  2  cwt.  25  lb.  ? 

16.  A  man  had  5  acres  of  land  which  he  divided  into  12 
equal  parts.     How  much  land  did  each  part  contain  ? 

17.  Divide  102  T.  15  cwt.  27  lb.  9  oz.  by  8. 

18.  Divide  16  bu.  3  pk.  6  qt.  by  2  bu.  1  pk. 

19.  I  have  84  lb.  14  oz.  of  salt  which  I  wish  to  put  into 
packages  of  2  lb.  6  oz.  each.     How  many  packages  will 
there  be? 

20.  If  a  horse  eats  1  pk.  2  qt.  of  oats  a  day,  how  many 
days  will  16  bu.  3  pk.  6  qt.  last  him  ? 

21.  How  many  sacks,  each  containing  2  bu.  3  pk.  2  qt., 
will  be  needed  to  hold  165  bu.  2  pk.  of  meal  ? 

22.  16  cwt.  75  lb.  9  oz.  of  butter  are  to  be  put -into  jars 
each  containing  9|  lb.     How  many  jars  will  be  needed  ? 

To  multiply  or  divide  a  compound  number  by  a  fraction. 

NOTE.  —  To  multiply  by  a  fraction,  multiply  by  the  numerator, 
and  divide  the  product  by  the  denominator. 

To  divide  by  a  fraction,  multiply  by  the  denominator,  and  divide 
the  product  by  the  numerator. 


68  SENIOR    ARITHMETIC. 

23.  How  much  is  f  of  16  hr.  17  min.  14  sec.  ? 

24.  How  much  is  £  of  30|  5^  19  8  gr.  ? 

25.  Divide  120  cd.  50  cd.  ft.  34  cti.  in.  by  |. 

26.  How  many  times  is  ji  contained  in  840  T.  15  cwt. 
98  Ib.  3  oz.  ? 

27.  A  man  sold  4  bu.  3  pk.  2  qt.  of  potatoes,  which  was 
f  of  what  he  raised ;  how  much  did  he  raise  ? 

28.  A  butcher  sells  120  tons,  9  cwt.  75  Ib.  of  beef  every 
month.     How  much  does  he  sell  in  §  of  a  month  ? 

29.  If  6  bottles  hold  5  gal.  2  qt.  of  milk,  how  much  milk 
will  3  such  bottles  hold  ? 

30.  A  field  contains  10  acres  12  sq.  rd.  of  land,  which 
is  |  the  size  of  the  whole  farm.      Find   the  size  of  the 
farm. 


fl.  A  railroad  track  extends  144  miles,  40  rd.  3  yd. 
How  far  has  a  train  of  cars  gone  which  has  travelled  T9^  of 
this  distance  ? 

32.  If  a  pipe  discharges  25  gal.  3  qt.  1  pt.  of  water  in 
1  hr.,  how  much  will  it  discharge  in  5|  hr.,  if  the  water 
flows  with  the  same  velocity  ? 

NOTE.  —  When  the  multiplier  or  divisor  is  a  mixed  number,  reduce 
to  an  improper  fraction,  and  proceed  as  above. 

33.  Divide  8  Ib.  11  oz.  15  pwt.  18  gr.  by  2§. 

34.  If  a  railroad  train  runs  60  mi.  35  rd.  16  ft.  in  one 
hour,  how  far  will  it  run  in  12f  hr.  at  the  same  rate  of 
speed  ? 

35.  Divide  14  bu.  3  pk.  6  qt.  1  pt.  by  £. 

36.  Divide  5  yr.  1  mo.  1  wk.  1  da.  1  hr.  1  min.  1  sec.  by 

3*. 


MISCELLANEOUS    PROBLEMS.  69 

MIS  CELL  A  NE  O  US   PR  OBLE  ^  I S. 

161.    1.    Xame  two  numbers  which  multiplied  together 
make  14. 

2.  Write  three  sets  of  factors  for  24. 

3.  Find  the  prime  factors  of  2205. 

4    Ux$X3  X2X$  _.> 
5  X  6  X  2  X  9  X  24 

5.  How  many  yards  of  silk,  24  inches  wide,  will  it  take 
to  line  a  skirt  containing  six  yards  of  cloth  28  inches  wide  ? 

6.  Find  the  least  common  multiple  of  2,  3,  4,  5,  6,  7, 

8,  9. 

7.  Find  the  greatest  common  divisor  of  285,  465. 

8.  Find  the  smallest  number  that  will  exactly  contain 

9,  15,  18,  20. 

9.  Find  the  length  of  the  longest  stick  that  will  exactly 
measure  the  sides  of  a  room  216  yd.  by  111  yd. 

10.  What  is  the  smallest  sum  of  money  with  which  you 
can  buy  pears  at  75/  a  basket,  peaches  at  90/,  and  grapes 
at  50/,  using  the  same  amount  of  money  for  each  kind  ? 

11.  How  many  times  is  1  contained  in  1  ? 

12.  How  many  times  is  1  contained  in  1  ? 

13.  A  man  bought  a  horse  for  $240,  which  is  f  of  what 
he  sold  it  for.     What  did  it  sell  for  ? 

14.  A  man  bought  a  horse  for  $240,  and  sold  it  for  f  of 
what  he  paid  for  it.     What  did  it  sell  for  ? 

15.  A  pole  stands  ^  in  the  mud,  1  in  the  water,  and  the 
remaining  10  feet  are  above  the  water.     How  long  is  the 
pole? 

16.  A  man  owns  4  farms  containing  365^,  375§,  284f. 
and  2544  acres  respectively.     How  many  acres  in  all  ? 


70  SIvNIOK    AK1THMET1C. 

17.  The  man  owning  the  above  farms  sells  A  234^  acres, 
and  B  366^\  acres.     How  many  acres  has  he  left  ? 

18.  What  is  the  value  of  3f  X  W  X  f  X  14  X  3T9o  X  ^ 
X  T62  X  |  X  T23  X  J. 

19.  Find  the  least  common  multiple  of  273,  462,  1785, 
and  399. 

20.  A  man  owns  a  farm  containing  400  acres.     He  sells 
J   of  the  farm,  and  divides  the  remainder  among  his  six 
children.     How  many  acres  does  each  child  receive  ? 

21.  Find  the  sum  of  3  bu.  6  pk.  2  qt.  1  pt.,  3  pk.  1  qt. 
1  pt.,  7  bu.  3  pk.,  4  bu.  7  qt.  1  pt.,  and  19  bu.  2  pk.  2  qt. 
1  pt. 

22.  Find  the  sum  of  4  Ib.  6  oz.  21  pwt.  9  gr.,  5  oz.  11  gr., 
3  Ib.  9  oz.  18  pwt.,  11  oz.  17  pwt.  5  gr.,  16  Ib.  4  oz.  11  pwt., 
18  Ib.  17  gr.,  21  Ib.  15  pwt.  11  gr.,  23  Ib.  10  oz.  21  pwt. 
23  gr. 

23.  What  part  of  a  mile  is  214  rd.  2  yd.  2  ft.  3  in.  ? 

24.  Reduce  1  pk.  4  qt.  If  pt.  to  the  fraction  of  a  bushel. 

25.  What  is  the  quotient  of  184  bales,  4  bundles,  1  ream, 
13  quires,  20  sheets,  divided  by  §  ? 

26.  A  farm  is  60  ch.  25  1.  long.     How  many  rods  long  is 
it? 

27.  A  surveyor  measured  my  farm,  and  found  that  it  is 
80  ch.  long  and  60  ch.  broad.     How  many  acres  does  it 
contain  ? 

28.  In  133128  in.  how  many  miles  ? 

29.  How  many  times  will  a  wheel  12  ft.  3  in.  in  circum- 
ference turn  round  in  going  15  mi.  20  rd.  12  ft.  2  in.  ? 

30.  How  many  steel  rails  30  ft.  long  are  needed  in  the 
construction  of  7  mi.  305  rd.  7  ft.  6  in.  of  double-track  rail- 
road ? 


MISCELLANEOUS   PKOBLEMS.  71 

31.  What  fraction  of  the  year  is  contained  in  the  months 
of  July  and  August,  1896  ? 

32.  The  greatest  depth  of  the  Atlantic  telegraph  cable  is 
2  mi.  250  rd.  5  yd.  1  ft.     How  many  feet  is  it  ? 

33.  How  many  statute  miles  in  45°  22'  30",  measured 
on  the  equator  ? 

34.  On  an  average  when  walking.  Isaac  steps  24  inches 
twice  every  second.     How  many  minutes  will  it  take  him 
to  Avalk  li  miles  ? 

35.  Reduce  ^  of  an  acre  to  square  rods  and  decimals  of 
a  square  rod. 

36.  A  silversmith  in  making  spoons  uses  2  Ib.  3  oz.  19 
pwt.  of  silver  in  one  day,  3  Ib.  18  pwt.  20  grs.  on  the  sec- 
ond day,  and  11  oz.  19  pwt.  23  gr.  on  the  third  day.     How 
much  silver  does  he  use  altogether  ? 

37.  What  decimal  of  a  pound  Troy  are  4  oz.  14  pwt.  ? 

38.  Reduce  18  pwt.  164  gr.  to  the  fraction  of  a  pound. 

39.  Find  the  difference'  between  f  Ib.  and  4  Ib.  7.84  oz. 
Troy. 

40.  Find  the  cost  of  2^  doz.  spoons,  each  weighing  9  oz. 
8.76  pwt.,  at  $.045  a  pennyweight. 

41.  What  decimal  part  of  a  grain  is  75*15-  of  a  pound? 

42.  What  part  of  an  acre  is  f  rd.  ? 

43.  Multiply  16  bu.  9  pk.  8  qt.  by  f . 

44.  What  is  the  product  of  20  bu.  9  oz.  11  pwt.  15.  gr. 
multiplied  by  f  ? 


72 


SEN10K   ARITHMETIC. 


ME  A  S  UREMENTS. 

162.  An   Angle    is    the    difference   in    direction    of    two 
lilies. 

163.  A  Right  Angle  is  the  angle  of 
a  square. 

164.  Anything  that  has  length  and 
breadth,  but  not  thickness,  is  a  Surface. 

165.  A  surface  that  does  not  change 
its  direction  is  a  Plane  Surface. 

166.  A  figure  having  four    straight 
sides  and  four  right  angles  is  a  Rect- 
angle. 

167.  A  Square  is  a  rectangle  having 

equal  sides. 


A  Right  Angle. 


A  Rectangle. 


168.  A  Square  Foot  is  a  square  1 
foot  long  and  1  foot  wide. 

169.  A    Square  Yard   is  a  square 
1   yd.   long  and  1  yd.  wide. 

170.  The  Area  of  a  surface  is  the 
number  of  square  units  that  it  con- 
tains. 


A  Square. 

NOTE.  —  There  are  6  sq.  in. 
in  a  row,  and  in  4  rows  there 
are  4  times  6  sq.  in.=  24  sq.  in. 

The    multiplier  is   abstract, 

and  the  unit  of  the  product  must  be  the  same  as  the  unit  of  the 
multiplicand. 


MEASUREMENTS. 


73 


171.  The  length  and  breadth  of  a  rectangle  are  called 
its  Dimensions. 

Length  X  Breadth  ==  Area. 
Area  -r-  Length        =  Breadth. 
Area  -7-  Breadth      =  Length. 

172.  A  figure    having  three    straight    sides    and    three 
angles  is  a  Triangle. 

The  Base  of  a  triangle  is  the  line  upon  which  it  stands, 
and  the  Altitude  is  its  height  above  the  base,  or  the  base 
extended.  Thus,  AC  is  the  base,  and  BD  the  altitude,  of 
the  triangle  shown  below. 


D 


r 


/         \ 


173.  It  is  evident  from  the  accompanying  figure  that  the 
area  of  a  triangle  is  equal  to  one-half  the  area  of  a  rec- 
tangle of  the  same  base  and  altitude. 

Every  circle  may  be  regarded  as 
composed  of  many  equal  triangles,  the 
radius  of  the  circle  forming  the  alti- 
tudes, and  the  circumference  forming 
the  sum  of  the  bases.  Therefore,  the 
area  of  a  circle  is  equal  to  1  the  prod- 
uct of  the  circumference  and  radius. 

PRINCIPLE. — The  circumference   of   a  circle  is  3.1416 
'   times  the  diameter,  or  about  3^. 

174.  Circumference  -?-  3.1416  =  Diameter. 
Diameter  x  3.1416  =  Circumference. 


74  SENIOR    ARITHMETIC. 

Oral. 

Find  the  areas  of  rectangles  as  follows : 

1.  10ft.  by  8  ft.  4.    50ft.  by  «20  ft. 

2.  16  ft.  by  4i  ft.  5.    6  ch.  by  8  ch. 

.  3.    14  rods  by  10  rd.  6.    9  yd.  by  6  yd. 

Find  the  other  dimension. 

7.  Area  24  sq.  ft.,  length  8  ft. 

8.  Area  72  sq.  yd.,  length  8  yd. 

9.  Area  100  sq.  in.,  breadth  5  in.1 

10.  Length  16  yd.,  area  64  sq.  yd 

11.  Breadth  4  ft.,  area  84  sq.  ft. 

12.  Area  56  sq.  ch.,  length  8  ch. 

Find  the  areas  of  the  following  triangles  : 

13.  Base  10  ft.,  alt,  12  ft.         16.    Base  5  ft.,  alt.  10  ft. 
y/14.    Base  9  yd.,  alt.  6  yd.          17.    Base  12  in.,  alt.  8  in. 

15.    Base  15  in.,  alt.  6  in.         18.    Base  10  rd.,  alt.  5£  rd. 

Find  circumferences  of  circles  having  the  following  di- 
ameters : 

NOTE.  —  Indicate  the  operation  only. 

19.  12  ft.         21.    16  in.          23.    16  rd.        25.    62  yd. 

20.  18  ft.         22.    14  yd.         24.    25  ch.        26.    84  ft. 

Find  the  diameters  having  the  following  circumferences  : 

NOTE.  —  Indicate  only. 

27.    78  ft.     28.    19  ft.     29.    316.14  rd.     30.    189.68  ch. 

Written. 
Find  areas. 

31.  Circumference  37.6992  ft.,  radius  6  ft. 

32.  Circumference  47.124  ft.,  diameter  15  ft. 


MEASUREMENTS.  75 

33.  Circumference  62.832  ft.,  diameter  20  ft. 

34.  Circumference  94.248  ft.,  diameter  30  ft. 

35.  Diameter  24  ft.  38.    Diameter  160  ft. 

36.  Radius  16  ft.  39.    Radius  62  ft. 

37.  Circumference  50  ft.     40.    Circumference  314.16  ft. 

41.  How  many  square  yards  are  there  in  a  floor  24  ft. 
long  and  15  ft.  wide  ? 

42.  The  base  of  a  triangle  is  20  ft.  and  the  altitude  18 
ft.     What  is  the  area  ? 

43.  The  circumference  of  a  circle  is  31.416  ft.  and  its 
radius  is  5  ft.     What  is  its  area  ? 

44.  When  the   diameter  of  a  circle  is  50  ft.,  what  is 
the  circumference  ? 

45.  When  the  radius  is  6  ft.,  what  is  the  circumference  ? 

46.  When  the  circumference   is  78.54   ft.,  what  is  the 
radius  ? 

^47.    Mr.   Clark's  farm  is  35  ch.  long  and  25  ch.  wide. 
How  many  acres  does  it  contain  ? 

48.  A  certain  field  is  70  rd.  long  and  65  rd.  wide.     How 
many  acres  are  there  in  the  field  ? 

49.  A  piece  of  land  is  65  rods  wide.     How  long  must 
it  be  to  contain  56 £  acres  ? 

o 

50.  How  many  sods  10  inches  square  will  be  required  to 
turf  a  lawn  100  ft.  long  and  50  ft,  6  in.  wide  ? 

51.  A  building  lot  measures   60  feet  in  front.     What 
must  be  its  depth  to  contain  1  of  an  acre  ? 

52.  How    many   tiles,    each   8    inches    square,    will   be 
required  for  the  floor  of  a  room  24  ft.  by  30  ft.  ? 

53.  How  many  shingles  will  be  required  for  a  roof  45 
ft.  long,  and  each  of  its  two  sides  20  ft.  wide,  allowing  8 
shingles  to  the  square  foot  ? 


76  SENIOR    ARITHMETIC. 

54.  How  many  square  yards  of  oil-cloth  are  needed  to 
cover  a  floor  18  ft.  by  24  ft.  6  in.  ? 

55.  A  owns  a  city  lot  168  ft.  long  and  42  ft.  wide.     He 
uses  |  of  it  for  a  lawn.     How  many  square  yards  does  the 
lawn  contain  ? 

56.  How  long  will  it  take  a  man  to  mow  the  above  lot, 
if  it  takes  him  a  minute  to  run  a  2-foot  lawn-mower  length- 
wise of  the  lot  ? 

57.  A  pony  can  reach  40  feet  in  any  direction  from  the 
stake  to  which  he  is   picketed.     Over  how  many  square 
rods  of  surface  can  he  graze  ? 

58.  What  is  the  diameter  of  a  tree  that  is  10  feet  in 
circumference  ? 

59.  A  basin   measures  9  inches   across  the  top  and  6 
inches  across  the  bottom.     How  much  farther  around  the 
top  than  around  the  bottom  ? 

60.  How  many  acres  of  land  are  enclosed  by  a  circular 
mile  track  ? 

61.  A  landscape  gardener  lays  off  a  circular  grass-plot 
whose  radius  is  one  rod,  and  near  it  a  semicircular  plot 
having  a  radius  two  rods  in  length.     Compare  their  areas. 

62.  If  the  area  of  a  triangle  is  9  acres  and  the  base  is 
80  rods,  what  is  the  altitude  ? 

63.  Find  the  area  of  one  gable  end  of  a  building  40  ft. 
wide,  the  ridge  being  15  ft.  above  the  eaves. 

64.  Find  area  of  a  triangle  whose  altitude  is  14  ft.  and 
its  base  12  ft. 

65.  Find  the  area  of  triangle  whose  altitude  is  4  in.  and 
base  12  in, 


MEASUREMENTS.  77 

CARPETING   ROOMS. 

175.  In  making  a  carpet,  the  carpeting  is  cut  from  a 
roll  into  strips  which  are  usually  laid  from  end  to  end 
of  the  floor,  or  lengthwise.  Sometimes  the  strips  are  laid 
across  the  room. 

1.  How  much  carpeting  must  I    purchase   to   cover  a 
room  6  yd.  long  and  4|  yd.  wide,  strips  running  length- 
wise ? 

SOLUTION.  —  It  will  be  necessary  to  purchase  as  much  carpeting 
as  if  the  room  were  5  yd.  wide,  the  excess  of  £  yd.  being  turned 
under  in  the  last  strip. 

1  strip  contains  6  yd.     5  strips  =  5  times  6  yd.  =  30  yd.     Ans. 

2.  How  many  yards  must        t i 

I  purchase,  if  the  strips  are 

laid  across  the  room  ? 


SOLUTION.  —  1  strip  contains 
4|  yd.  6  strips  =  6  times  4f  yd. 
=  28^  yd.  Ans. 

Carpeting  is  commonly  1 
yd.  or  |  yd.  wide. 


NOTE.  —  It  is  often  necessary  ^  ^ 

to  purchase  more  than  enough 

carpeting  to  cover  a  room,  on  account  of  the  waste  in  matching 
patterns. 

This  diagram  represents  the  floor  in  Ex.  1.  in  which  the 
strips  are  laid  lengthwise.  Pupils  should  draw  a  similar 
diagram  for  each  floor. 

NOTE.  —  Carpeting  is  sold  by  lineal  yards  or  meters,  not  by 
square  measure. 

3.  A  merchant  bought  a  roll  of  carpet  containing  74  yd. 
at  85/  a  yard,  and  sold  it  at  $1.15  a  yard.  What  was  his 
profit  ? 


78  SENIOR    ARITHMETIC. 

4.  A  roll  of  carpet  |  yd.  wide  contains  60  yards ;  how 
many  square  yards  of  surface  will  it  cover  ? 

5.  How  many  strips  of  carpet  3  ft.  wide  will  cover  a 
floor  15  ft.  wide  ?     17  ft  wide  ?     18  ft.  wide  ? 

6.  How  many  strips  27  in.  wide  are  required  for  a  floor 
12  ft.  wide  ?     14  ft.  wide  ?     16  ft.  wide  ? 

7.  If,  in  Example  5,  the  room  is  16  ft.  long,  how  many 
linear  yards  of  carpet  will  be  needed  to  cover  the  floor  ? 

8.  If,  in  Example  6,  the  room  is  19  ft.  long,  how  many 
yards  will  be  required  to  cover  the  floor  ? 

9.  How  many  yards  of  ingrain  carpet  f  yd.  wide  will  be 
•required  for  a  floor  17  ft.  wide  and  20  ft.  long,  strips  run- 
ning across  the  room  ? 

10.  How  much  will  a  carpet  cost  at  f  .90  a  yard  to 
cover  a  floor  22  ft.  long  and  15  ft.  wide,  if  the  strips  run 
crosswise,  and  no  allowance  is  made  for  matching  ? 

11.  How  many  yards  of  carpet  f  yd.  wide  will  be  re- 
quired for  a  floor  20  ft.  long  and  15  ft.  wide,  if  the  strips 
run  across  the  room  ? 

12.  How  many  yards  of  carpet  1  yd.  wide  will  be  re- 
quired for  a  floor  18  ft.  long  and  14  ft.  wide,  strips  running 
across  the  room  ? 

13.  How  many  yards  of  carpeting  are  needed  to  cover 
a  floor  24  *ft.  long  and  17  ft.  wide,  strips  running  length- 
wise and  |  yd.  wide  ? 

14.  If  my  room  is  16^  ft.  long  and  12  ft.  wide,  how 
many  yards  of  carpeting  24  inches  wide  must  I  buy,  if  in 
cutting  6  inches  is  allowed  on  each  strip  for  matching  ? 

15.  I  wish  to  have  a  carpet  woven.     My  room  is  21  ft. 
long  and  17  ft.  wide ;  how  much  carpeting,  34  inches  wide, 
must  I  order  to  exactly  cover  the  room,  no  allowance  being 
made  for  matching  ? 


MEASUREMENTS.  79 

16.  How  many  yards  of  carpet  2^  ft.  wide  will  cover  a 
floor  7£  yards  long  and  14  ft.  wide,  if  strips  run  length- 
wise, and  it  requires  ^  yd.  for  matching  ? 

17.  What  will  it  cost  to  carpet  a  room  15  ft.  by  17^  ft., 
with  carpet  30  inches  wide,  at  $1.20  per  lineal  yard,  if  the 
strips  run  lengthwise,  and  an  allowance  of  9  in.  to  each 
strip  be  made  for  matching  ? 

18.  How  much  less  would  be  the  cost  with  no  loss  for 
matching  ? 

19.  A  room  31  ft.  by  17  ft.  is  to  be  covered  with  carpet- 
ing 30  in.  wide.    How  many  yards  must  be  purchased,  and 
how  wide  a  strip  must  be  turned  under  ? 

20.  At  $2.50  a  yard,  what  will  be  the  cost  of  a  carpet 
to  cover  a  parlor  floor  6  yd.  long  and  5±  yd.  wide,  if  f  yd. 
is  wasted  in  matching  ? 

21.  How  many  yards  of  matting  1^  yd.  wide  will  be 
required  for  an  assembly  room  85  ft.  8  in.  long  and  64  ft. 
6  in.  wide,  strips  running  across  the  room  ? 

22.  A  room  17  ft.  6  in.  long,  14  ft.  wide,  is  to  be  car- 
peted with  carpet  f  yd.  wide.     A  border  f  yd.  wide  goes 
around  the  outside.     How  many  yards  of  border,  and  how 
many  yards  of  carpet  if  strips  run  lengthwise,  and  there 
is  a  waste  of  one  foot  on  each  strip  for  matching  ? 

PLASTERING  AND   PAINTING,    ETC. 

176.  Plastering  and  Painting  are  usually  done  by  the 
square  yard.  Allowance  is  sometimes  made  for  doors  and 
windows,  which  are  called  openings.  Allowance  is  also 
sometimes  made  for  base-boards  and  wainscoting. 

1.  A  room  is  18  ft.  long,  12  ft.  wide,  and  10  ft.  high. 
How  many  square  yards  in  the  walls  and  ceiling,  making 
no  allowance  for  openings  ? 


80 


SENIOR   ARITHMETIC. 


12  ft. 


NOTE.  —  Let  the  pupils  draw  a  diagram  for  each  room,  represent- 
ing the   four  walls  in  a  line.      The  entire 
length   of  the   walls  will   be  2  x  (18  ft.  4 
12  ft.)  =  60  ft.     The  area  of  the  four  walls, 
60  ft.  by  10  ft.  =  600  sq.  ft. 

Area  of  ceiling  18  ft  by  12  ft.  =  216  sq.  ft. 
600  sq.  ft.  +  216  sq.  ft.  =  816  sq.  ft.  =  77§      - 
sq.  yd.,  area  of  walls  and  ceiling. 


18  it. 


12  ft. 


18  ft. 


12ft. 


2.  Find  the  cost  of  plastering  the  walls  and  ceiling  of 
a  room  35  ft.  long,  26  ft.  6  in.  wide,  and  15  feet  high,  at 
$ .45  a  sq.  yd.,  allowing  1024  sq.  ft.  for  doors,  windows,  and 
base-board. 

3.  How  many  square  yards  of  plaster  in  the  sides  and 
ceiling  of  a  room  30  ft.  long,  24  ft.  wide,  and  10  ft.  high, 
allowing  for  a  base-board  1  ft.  high,  2  doors  3  ft.  by  8  ft., 
and  4  windows  3  ft.  by  6  f t.  ? 

4.  Find  the  cost  of  plastering  the  ceiling  of  a  room  18 
ft.  by  20  ft.,  at  10  cents  a  square  yard. 

5.  A  room  15  ft.  by  18  ft.,  and  10  ft.  high,  has  4  doors 
each  3  ft.  by  7  ft.,  and  three  windows  each  3  ft.  by  6  ft. 
Find  the  cost  of  plastering  the  walls  and  ceiling  of  the 
room  at  30  cents  a  square  yard,  deducting   one-half   the 
surface  for  openings. 

6.  How  many  sq.  yds.  of  plastering  in  the  ceiling  of 
a  room  20  ft.  long,  9  ft.  high,  and  15  ft.  wide,  no  allowance 
for  openings  ? 

7.  At  $.30  a  sq.  yd.,  how  much  will  it  cost  to  plaster 
a  room  21  ft.  6  in.  long,  16  ft.  wide,  and  9  ft.  high,  the 
base-board  being  8  inches  wide,  and  allowing  for  3  windows 


MEASUREMENTS.  81 

7  ft.  by  2i  ft.,  and  six  doors  of  the  same  dimensions  as 
the  windows  ? 

8.  Find  the  cost  of  plastering  the  walls  and  ceiling  of 
a  room  which  is  36  ft.  long,  27  ft.  wide,  and  9  ft.  high,  at 
25^  per  square  yard. 

9.  My  study  is  18  ft.  long,  16  ft.  wide,  8^  ft.  high,  and 
contains  1  door  3  ft.  by  7  ft.,  and  2  windows,  each  3  ft.  by 
6  ft.     The  base-board  is  9  in.  high.     What  will  it  cost, 
at  36/  per  sq.  yd.,  to  plaster  it,  making  full  deduction  for 
openings  ? 

10.  At  35  cents  a  sq.  yard,  what  will  be  the  cost  of 
plastering  the  walls  and  ceiling  of   a  room  6   yd.   long, 
5  yd.  wide,  and  3  yd.  high,  an  allowance  of  20  sq.  yards 
being  made  for  openings,  etc.  ? 

11.  Find  the  cost  of  plastering  a  room  18  ft.  square  and 
10  ft.  high,  at  25  cents  a  sq.  yard,  £  being  deducted  for 
openings  ? 

12.  A  close  fence  6^  ft.  high  surrounds  a  vacant  lot  450 
ft.  by  380  ft.     At  7  cents  a  sq.  yard,  what  will  be  the  cost 
of  painting  both  sides  of  the  fence  ? 

13.  Find  the  cost  at  18/  per  sq.  yard  to  plaster  the 
sides  and  bottom  of  a  cistern  8  ft.  6  in.  square,  and  9  ft. 
deep. 

14.  Find  the  square  yards  of  plastering  on  a  room  20  ft. 
long,  17  ft.  6  in.  wide,  9  ft.  high.     Allow  for  6  windows, 
each  7  ft.  6  in.  high,  3  ft.  wide,  and  4  doors,  each  7  ft. 
high  and  3  ft.  9  in.  wide. 

PAPERING  WALLS. 

177.  Wall-paper  is  sold  by  the  roll.  A  Single  Eoll  is 
8  yd.  long  ;  a  Double  Eoll,  16  yd.  long.  Borders  are  sold 
by  the  lineal  yard. 

The  number  of   rolls  needed  for  a  room  is  found  by 


82  SENIOR   ARITHMETIC. 

dividing  the  area  of  the  space  to  be  papered  by  the  area  of 
one  roll.     The  width  of  wall-paper  is  commonly  18  in. 

NOTES. — Unless  otherwise  stated,  a  roll  is  considered  as  8  yd. 
long  and  18  in.  wide. 

Dealers  in  wall-paper  do  not  sell  a  part  of  a  roll.  If  a  part  of  a 
roll  is  needed,  a  whole  roll  must  be  purchased. 

1.  What  would  it  cost  to  paper  the  walls  of  a  room  18 
ft.  long,  12  ft.  wide,  and  9  ft.  high,  with  paper  8  yd.  to 
the  roll  and  ^  yd.  wide,  at  45/  a  roll  ? 

2.  How  many  strips  of  paper,  and  how  many  double 
rolls,  will  paper  the  sides  of  a  room  15  ft.  long,  12  ft.  wide, 
and  8  ft.  high,  each  roll  being  1£  ft.  wide,  and  16  yd.  long, 
no  allowance  being  made  for  matching  ? 

3.  How  many  double  rolls  of  paper  16  yd.  to  the  roll, 
•J  yd.  wide,  will  be  required  to  paper  the  walls  and  ceiling 
of  a  room  25  ft.  long,  10  ft.  wide,  and  10  ft.  high,  110 
sq.  ft.  being  deducted  for  doors,  windows,  etc  ? 

4.  How  many  rolls  of  paper  will  be  required  for  the 
walls  of  a  room  16  ft.  by  20  ft.,  and  9  ft.  high  above  the 
base-board,  allowing  for  3  doors,  each  3  ft.  by  7  ft.,  and  3 
windows,  each  3  ft.  by  6  ft.  ? 

5.  What  will  be  the  cost  of  the  paper  and  border  for 
the  above  room  at  30  cents  a  roll  for  the  paper,  and  15 
cents  a  yard  for  the  border  ? 

6.  How  many  rolls   of   paper  must  be   purchased   to 
paper  the  walls  and  ceiling  of  a  library,  12  ft.  long,  10  ft. 
6  in.  wide,  and  8  ft.  high,  the  base-board  being  6  in.  wide,  and 
the  border  1-J-  ft.  wide,  with  paper  \  yd.  wide  and  8  yd.  long, 
the  paper  extending  from  the  border  to  the  base-board  ? 

BOARD   MEASURE. 

/ 178.    A  Board  Foot  is  a  square  foot  of  the  surface  of  a 
board,  1  inch  thick,  or  less. 


MEASUREMENTS.  83 

To  find  the  number  of  board  feet  in  lumber  that  is 
more  than  one  inch  thick,  we  must  multiply  the  num- 
ber of  board  feet  in  the  surface  by  the  number  of  inches 
in  the  thickness. 

A  board  10  ft.  long,  1  ft.  wide,  and  1  in.  thick  or  less  con- 
tains 10  board  feet ;  but  a  beam  10  ft.  long,  1  ft.  wide,  and 

8  in.  thick  contains  8  times  10  board  feet  =  80  board  feet. 
To  find  the  number  of  board  feet  in  a  tapering  board, 

the  average  width  must  be  found  by  taking  £  the  sum  of 
the  widths  of  the  two  ends.  Thus  a  board  10  ft.  long,  and 
12  in.  wide  at  one  end,  and  6  in.  wide  at  the  other,  con- 
tains as  many  board  feet  as  if  it  had  a  uniform  width  of 

9  inches.     (12  -f-  6)  -5-  2  =  9. 

179.  The  number  of  board  feet  =  Length  (in  feet)  x 
Width  (in  feet)  X  Thickness  (in  inches). 

NOTE.  —  When  the  thickness  is  one  inch  or  less,  the  number  of 
board  feet  is  the  product  of  the  length  and  width  in  feet. 

1.  How  many  board  feet  in  a  board  15  ft.  long,  15  in. 
wide,  and  1  inch  thick  ? 

2.  How  many  board  feet  would  there  be  in  the  board 
(Ex.  1)  if  it  were  f  in.  thick  ?     2  in.  thick  ?    1£  in.  thick  ? 

3.  How  many  feet  of  lumber  one  inch  thick  will  be 
required  for  a  tight  board  fence  6  ft.  high  around  a  yard 

4  rods  square  ? 

4.  How  much  lumber  (Ex.  3)  will  be  required  for  an 
open  board  fence,  4  boards  high,  boards  8  inches  wide,  and 

5  inches  apart  ? 

5.  I  need  213  planks  4  ft.  8  in.  long,  1  ft.  wide,  and  2 
inches  thick,  to  build  a  sidewalk.     How  much  will  they  cost 
at  $13  a  thousand  ? 

6.  How  many  feet  of  lumber  will  it  take  to  build  a  line 
fence  168  ft.  long,  the  fence  being  5  boards  high,  and  the 
boards  12  ft.  long  and  6  in.  wide  ? 


84  SENIOR   ARITHMETIC. 

7.  What  will  be  the  cost  of  10  planks  each  12  ft.  long, 
10  in.  wide,  and  3  in.  thick  at  $16  per  M.  ? 

8.  Find  the  cost  of  a  stick  of  timber  8  in.  square,  and 
40  ft.  long,  at  $18  per  M. 

9.  What  is  the  cost  of  8  sticks  of  timber  each  36  ft. 
long,  10  in.  wide,  8  in.  thick,  at  $12  per  M.  ? 

10.  How  many  board  feet  of  2-inch  planking  will  it  take 
to  make  a  box  2  ft.  8  in.  long,  2  ft.  wide,  1  ft.  6  in.  deep 
inside  ? 

11.  Find  the  cost  of  7  planks  12  ft.  long,  16  in.  wide  at 
one  end,  and  12  in.  at  the  other,  at  $.08  a  board  foot. 

12.  At  $18  per  M.,  find  the  cost  of  flooring  a  room  21 
ft.  by  16  ft.,  allowing  1  of  the  lumber  for  matching. 

NOTE.  —  Find  area  of  floor  and  add  &. 

13.  Find  the  cost  of  a  board  20  ft.  long,  22  in.  wide  at 
one  end,  and  tapering  to  16  in.  at  the  other,  and  l£  in. 
thick,  at  $30  per  M.  ? 

14.  At  $12  per  M.,  what  will'  be  the  cost  of  2-inch  plank 
for  a  3  ft.  6  in.  sidewalk  on  the  street  side  of  a  rectangular 
corner  lot  56  ft.  by  106  ft.  6  in.  ? 

MISCELLANEOUS. 

180.  1.  My  dining-room  is  15  ft.  long  and  12  ft.  wide ; 
the  walls  are  10  ft.  high.  What  will  it  cost  to  paper  the 
walls  and  ceiling  with  paper  l£  ft.  wide,  if  there  are  8  yd. 
in  a  roll,  and  each  roll  costs  $.37£.  (TL  allowed  for 
openings.) 

2.  What  will  a  carpet  for  the  dining-room  (Ex.  1)  cost 
me  at  $.75  a  yd.,  carpet  f  yd.  wide  ? 

3.  There  are  three  windows  in  the  dining-room.     What 
will  it  cost  to  furnish  them  with  shades  at  $1.10  each  and 
sash-curtains  at  $1.37^  each  ? 


MEASUREMENTS.  85 

4.  I  bought  a  table  at  $14.50,  six  chairs  at  $2.75  each, 
a  sideboard  for  $30,  and  other  furniture  for  $28.97.      I 
also  spent  $20  for  new  table  linen  ;  what  did  it  all  cost  ? 

5.  What  was  the  entire  cost  of  refurnishing  my  dining- 
room  ? 

6.  What  will  it  cost  to  carpet  a  room  which  is  24  ft. 
long,  and  18  ft.  wide,  with  Brussels  carpet  1  yd.  wide;  no 
waste  in  matching,  at  $1  per  yard  ? 

7.  Find  the  cost  of   plastering  sides  and  ceiling  of  a 
room   26  ft.  long,  13^   ft.   wide,  13  ft.   high,   at  9  cents 
a  square  yard,  allowing  25  sq.  yd.  for  openings. 

8.  What  will  it  cost  to  build  a  cement  walk  40  ft.  long, 
and  6  ft.  wide,  at  $1.25  per  sq.  yd.  ? 

9.  A  field,   containing  8  acres,  is  60  rd.  long.     How 
wide  is  it  ? 

10.  How  many  yards  of  carpet  will  cover  a  floor  18  ft. 
long,  16  ft.  wide  ?     Carpet  one  yard  wide,  strips  to  run 
lengthwise  of  room. 

How  many  yards  if   the    strips    run    crosswise   of   the 
room  ? 

11.  Find  the   cost  of  fencing  a  rectangular  corner  lot 
68  ft.  by  130  ft.,  the  street  fence  costing  54  cts.  a  yard, 
and  the  line  fences  25  cents  a  yard,  but  only  half  of  the 
cost  of  the  latter  to  be  charged  to  the  lot. 

12.  Find  the  cost  of  a  carpet  f  of  a  yd.  wide,  at  $1.50 
per  lineal  yard,  for  a  room  20  ft.  long  and  18  ft.  wide, 
strips  running  lengthwise,  and  allowing  a  waste  of  £  of  a 
yd.  on  each  strip  for  matching. 

13.  What  is  the  breadth  of  a  rectangular  lot  whose  area 
is  75  sq.  ch.,  and  the  length  9  ch.  ? 

14.  What  is  the  circumference  of  a  circle  whose  radius 
is  9  ft.  ? 


00  SENIOR   ARITHMETIC. 

15.  How  many  square  yards  in  the  above  circle  ? 

16.  How  many  revolutions  does  the  5-foot  driving-wheel 
of  a  locomotive  make  in  going  30  miles  ? 

17.  Find  the  area  of  a  triangle  whose  base  is  5  feet  and 
altitude  3  ft. 

18.  If  the  circumference  of  the  earth  is  25000  miles, 
what  is  the  diameter  ? 

19.  The  circumference  of  a  circle  is  18  ft.     What  is  its 
radius  ? 

20.  How  many  sq.  yards  in  a  triangle  whose  base  is  48 
ft.  and  whose  altitude  is  24  feet  ? 

21.  Find  the  area  of   the  gable-end  of   a  house  whose 
width  is  25  feet  and  whose  ridge  is  10  feet  6  inches  higher 
than  the  base  of  the  gable. 

22.  If  the  diameter  of  the  earth  is  8000  miles,  what  is 
the  circumference  ? 

23.  How  many  board  feet  in  10  planks  18  ft.  long,  15 
in.  wide,  and  2  in.  thick  ?  and  what  will  they  cost  at  $40 
per  M.  ? 

24.  Find  the  cost  of  10  joists,  3  in.  by  12  in.,  16  ft.  long, 
at  $25  per  M. 

VOLUMES. 

181.  Anything  that  has  length,  breadth,  and  thickness 
is  called  a  Solid  or  Volume. 

182.  A  Rectangular  Volume  is  a  solid  having  six  rectan- 
gular faces. 

183.  A  Cube  is  a  solid  having  six  equal  square  faces. 

184.  A  Cubic  Inch  is  a  cube  1  inch  long,  1  inch  wide,  and 

1  inch  thick. 

185.  The  Volume  or  Solidity  of  a  body  is  the  number  of 
cubic  units  that  it  contains. 


MEASUREMENTS. 


87 


3  dn.  wide 


1.    How  many  cubic  inches  in  a  block  4  in.  long,  3  in. 
wide,  and  2  in.  thick  ? 

The  block  is  made  up  of  two  layers,  each  \  in.  thick.     In  the  top 
layer  there  are  4  times  3  cu.  in.     In 
the  two  layers,   therefore,   there  are 
2  x  (4  x  3  cu.  in.)  =  24  cu.  inches. 

The  multiplier  must  be 
considered  as  abstract. 

The  three  dimensions 
must  have  the  same  unit. 
The  length,  breadth, 
and  thickness  of  a  rec- 
tangular solid  are  its  di- 
mensions. 


3  cu.  in. 
_4 

12  cu.  in. 

2 

24  cu.  in. 


186.    Length  x  breadth  x  thickness  =  Solidity. 
Solidity  -±  either  dimension  =  the  product  of  the  other 
two. 

Solidity  -f-  the  product  of  two  dimensions  =  the  other. 

2.  Find  the  number  of  cubic  feet  of  air  in  a  school- 
room 32  ft.  square  and  12  ft.  high. 

3.  How  high  is  a  room  that  is  24£  ft.  long,  20  ft  wide, 
and  contains  4410  cu.  ft.  ? 

4.  A  cu.  foot  of  ice  weighs  56J  pounds.      How  much 
will  a  load  of  22  cakes  weigh,  each  cake  measuring  2  ft. 
square,  and  1   ft.   thick  ? 

5.  The    capacity  of    a  rectangular  box   is  480   cu.  in 
The  box  is  8  in.  wide,  and  5  in.  deep.     How  long  is  it9 

6.  A  schoolroom  is  25  ft.  long,  18  ft.  wide,  and  12  ft. 
high.     If  60  pupils  are  seated  in  it,  how  many  cu.  ft.  of 
air  are  allowed  for  each  child  ? 

7.  A  man  sold  3  blocks  of  Vermont  marble,  each  8  ft. 
long,  and  6  in.  x  6  in.  at  the  ends.     How  much  did  he 
receive  for  the  marble  at  $3.50  per  -cu.  ft.  ? 


88  SENIOR    ARITHMETIC. 

8.  A  hot-house  bed  is  3  ft.  9  in.  long,  and  3  ft.  4  in. 
wide,   inside  measure.     How  deep  must  it  be  to  contain 
25  cu.  ft.  of  earth,  and  allow  6  in.  for  the  growth  of  the 
plants  ? 

9.  How  many  bricks  8  in.  by  4  in.  and  2  in.  thick  will 
be  needed  for  a  wall  60  ft.  long,  20  ft.  high,  and  2  ft. 
thick,   making  no  allowance  for  mortar  ? 

10.  How  many  rectangular  blocks  12  in.  by  8  in.  by 
in.  can  be  packed  into  a  wagon-box  10  ft.  long,  4  ft.  wide, 
and  2  ft.  6  in.  deep  ? 

11.  How  many  cubic  yards  of  earth  must  be  excavated 
from  a  cellar  30  ft.  10  in.  long,  21 1  ft.  wide,  and  5  ft.  8  in. 
deep  ? 

12.  How  many  square  feet  in  the  surface  of  a  rectangu- 
lar box  3  ft.  4  in.  long,  2  ft.  2  in.  wide,  and  1£  ft.  high  ? 

13.  How  many  cubes  2^   inches  on  each  edge  can  be 
sawed  from  a  block  of  marble  10  ft.  2J  in.  long,  6  ft.  5  in. 
wide,  and  3  ft.  4  in.  thick  ? 

14.  A  box  is  1.5  in.  long,  .85  in.  wide,  and  .58  in.  deep. 
What  is  its  capacity  in  cubic  inches  ? 

15.  The  altitude  of  a  cylinder  is  8  ft.  and  the  circum- 
ference of  the  base  is  3  ft.     What  are  the  cubic  contents 
of  the  cylinder  ? 

NOTES.  — Contents  of  a  cyclinder  =  Area  of  Base  x  Altitude. 

Area  of  curved  surface  of  a  cylinder  =  circumference 

of  Base  x  Altitude. 
This  may  be  seen  by  cutting  a  piece  of  paper  so  that   it  will 
exactly  cover  the  curved  surface  of  a  small  cylinder. 

16.  What  is  the  area  of  the  curved  surface  in  the  cylin- 
der mentioned  in  Ex.  15  ? 

17.  How  much  tin  will  be  required  to  make  2  doz.  cylin- 
drical shaped  cans,  with  a  diameter  of  4  in.  and  altitude 


MEASUREMENTS.  89 

of  7  in.,  allowing  tin  for  the  curved  surface  and  the  two 
circular  ends  ? 

18.  How  many  cubic  inches  of  water  will  the  2  doz.  cans 
(Ex.  17)  contain  ? 

19.  What  will  it  cost  to  dig  a  cellar  36  ft.  long,  24  ft. 
wide,  and  6  ft.  deep,  at  20  cts.  a  cubic  yard  ? 

WOOD   MEASURE. 

187.  A  pile  of  wood  8  feet  long,  4  feet  wide,  and  4  feet 
high  makes  a  Cord. 

One  of  the  8  feet  in  length  of  a  cord  of  wood  is  a  Cord 
Foot. 

NOTE.  —  This  may  be  illustrated  by  placing  side  by  side  8  books 
of  equal  size.  One  of  the  books  represents  a  cord  foot. 

How  many  cords  of  wood  in  the  following : 

1.  A  pile  18  ft.  long,  4  ft.  wide,  8  ft.  high  ? 

2.  A  pile  50  ft.  long,  8  ft.  wide,  6  ft.  high  ? 

3.  A  pile  19  ft.  long,  2  ft.  wide,  5£  ft.  high  ? 

4.  A  pile  16  ft.  long,  4-J  ft.  wide,  7  ft.  high  ? 

5.  What  is  the  cost  of  a  pile  of  wood  10  ft.  long,  4  ft. 
wide,  and  8  ft.  high,  at  $4J  a  cord  ? 

6.  How  high  must  a  pile  of  wood  be  piled  to  contain 
10  cords,  if  the  pile  is  50  ft.  long  ? 

7.  How  many  cords  of  wood  can  be  piled  in  a  shed 
24  ft.  long,  18  ft.  wide,  and  12  ft.  high  ? 

8.  How  many  cords  of  building-stone  in  a  pile  18  ft. 
long,  6£  ft.  wide,  and  3  ft.  high  ? 

9.  At  $3.50  a  cord,  what  will  be  the  cost  of  a  pile  of 
stone  15  ft.  long,  4|  ft.  wide,  and  5  ft.  high  ? 

10.  How  many  cubic  feet  in  a  cord  of  2-foot  wood  ? 
3-foot  wood  ?  18-inch  wood  ? 


90  SENIOR   ARITHMETIC. 

CAPACITY   OF  BINS. 

188.  1.  A  bushel  fills  2150.42  cubic  inches  of  space. 
How  many  bushels  of  wheat  can  be  contained  in  a  bin  5 
ft.  X  5  ft.  X  4  ft.  ? 

5  x  5  X  4  x  1728  -s-  2150.42. 

NOTE.  — A  bushel  fills  1£  cu.  ft.  of  space  nearly. 

2.  A  wine  gallon  fills  231  cubic  inches  of  space.     How 
many  gallons  of  water  can  be  contained  in  a  rectangular 
tank  10  ft.  by  8  ft.  by  4  ft.  ? 

NOTE. — A  cubic  foot  of  space  contains  ^/f-  gal.  =  7&  gal. 
nearly. 

Find  the  contents  in  bushels: 

3.  Of  a  bin  6  ft.  long,  5  ft.  wide,  and  4  ft.  high. 

4.  Of  a  wagon-box  10  ft.  long,  42  in.  wide,  and  22  in. 
high. 

5.  Of  a  box  3  ft.  by  2£  ft.  by  2£  ft. 

6.  How  high  must  a  bin  8  ft.  long  and  5  ft.  wide  be 
built  to  contain  120  bushels  ? 

Find  the  contents  in  gallons. 

7.  Of  a  tank  8  ft.  by  6  ft.  by  2£  ft. 

8.  Of  a  cistern  6  ft.  by  5  ft.  by  4£  ft. 

9.  Of  a  tank  5£  ft.  square  and  6  ft.  deep. 

10.  How  many  barrels  of  water  will  a  cistern  contain 
that  is  6  ft.  by  6  ft.  by  7  ft.  ? 

11.  A  circular  cistern  is  5  ft.  in  diameter  and  6  ft.  deep. 
How  many  barrels  of  water  will  it  hold  ? 

NOTE.  — Area  of  base  x  altitude. 

12.  How  deep  must  I  build  a  bin  that  is  6  ft.  square,  to 
hold  90  bushels  of  wheat  ? 

13.  How  deep  must  I  build  a  tank  that  is  5  ft.  square 
to  hold  40  barrels  ? 


LONGITUDE   AND   TIME.  V     91j 


LONGITUDE  AND   TIME. 

189.  A  Meridian  is  an  imaginary  line  running  from  the 
north  pole  to  the  south  pole. 

All  places  on  a  meridian  have  the  same  time. 

NOTE.  —  The  meridians  of  Greenwich  and  Washington  are  the 
meridians  that  run  through  Greenwich  and  Washington. 

190.  Longitude  is  distance  east  or  west  from  some  stan- 
dard meridian,  as  Greenwich  or  Washington.     When  two 
places  are  on  the  same  side  of  the  standard  meridian,  their 
difference  in  longitude  is  found  by  subtraction.     When  011 
opposite  sides,  their  difference   in   longitude  is  found  by 
addition. 

1.  What  is  the  difference  in  longitude  between  two 
cities,  one  of  which  is  20°  west  longitude,  the  other  30° 
east  longitude 


20o  +  30o  =  50o. 

2.  What  is  the  difference  in  longitude  between  two 
places,  one  of  which  is  40°  E.,  the  other  70°  E.  ? 

70°  _  40°  =  30°.     Ans. 

NOTE.  —  No  two  places  can  have  a  difference  in  longitude  exceed- 
ing 180°.  If,  in  finding  difference  in  longitude  by  addition,  the  sum 
exceeds  180°,  subtract  the  sum  from  360°  to  find  the  true  difference. 

The  earth  turns  upon  its  axis  from  west  to  east  once  in 
24  hours,  thus  ^  of  its  entire  circumference,  360°,  or  15° 
of  longitude,  passes  under  the  sun  in  1  hour. 

Since  the  earth  turns  at  the  rate  of  15°  every  hour,  in 
1  minute  it  turns  -^  of  15°,  or  15',  and  in  1  second  ^  of 
15',  or  15".  Hence, 

The  earth  rotates  15°  in  1  hour,  15'  in  1  minute,  and 
15"  in  1  second. 

SS.    The  difference  in  longitude  between  two  cities  is  18°, 
30'.     What  is  the  difference  in  time  ? 


92  SENIOR   ARITHMETIC. 

SOLUTION.  —  Since  the  earth  turns  15° 

15  /18°    3CK in  1  hr.,  15'  in  1  min.,  and  15"  in  1  sec.,  the 

1  hr.   14  iniD.      time  can  be  found  by  dividing  the  number 
of  degrees,  minutes,  and  seconds  by  15. 

4.  The  difference  in  time  between  two  cities  is  54  inin. 
19  sec.  What  is  their  difference  in  longitude  ? 

54  min.  19  sec.          Since  the  earth  turns  15°  in  1  hr.,  15'  in 
15  1  min.,  and  15"  in  1  sec.,  the  distance  in 

13°  34'  45"  degrees,  minutes,  and  seconds  may  be  found 

by  multiplying  the  number  of  hours,  min- 
utes, and  seconds  by  15. 

STANDARD   OR  RAILROAD   TIME. 

191.  The  railroad  companies  have  divided  the  country 
into  four  time  belts,  extending  north  and  south.  All 
places  in  each  belt  take  the  time  of  the  meridian  which 
passes  through  or  near  the  middle  of  the  belt.  The  belts 
are  as  follows :  Eastern,  Central,  Mountain,  and  Pacific. 

The  standard  meridian  for  the  Eastern  belt  is  the  75th, 
for  the  Central  belt  the  90th,  for  the  Mountain  belt  the 
105th,  and  for  the  Pacific  belt  the  120th. 

These  standard  meridians  are  15  degrees  apart. 

Therefore,  when  it  is  noon  in  the  Eastern  belt  it  is  11 
A.M.  in  the  Central  belt,  10  A.M.  in  the  Mountain  belt,  and 
9  A.M.  in  the  Pacific  belt. 

In  going  westward  into  another  time  belt,  the  traveller 
sets  his  watch  back  one  hour. 

In  travelling  eastward,  he  sets  his  watch  ahead  one  hour. 

When  it  is  noon  on  the  standard  meridian  of  each  belt, 
it  is  called  noon  at  all  places  in  the  belt. 

NOTE.  —  Time  reckoned  by  this  method  is  not  true  solar  time, 
but  it  secures  a  uniformity  of  time  which  is  very  desirable. 

The  time  in  general  use  is  Railroad  or  Standard  Time. 


LONGITUDE   AND   TIME.  93 

Oral. 

5.  When  it  is  5  P.M.  Mountain  time,  what  is  the  time 
in  the  Pacific  belt  ? 

6.  When  it  is  11  A.M.  Pacific  time,  what  is  the  Central 
time? 

7.  In  travelling  from  San  Francisco  to  New  York,  how 
many  times  do  I  change  my  watch  ?  and  do  I  set  it  ahead 
or  back  ? 

8.  When  it  is  4  A.M.  at  Augusta,  Me.,  what  is  the  stan- 
dard time  at  St.  Louis  ? 

9.  When  it  is  1  P.M.  Mountain  time  at  Denver,  what 
time  is  it  at  Washington,  D.C.  ? 

10.  AVhat  is  the  Pacific  time  at  San  Francisco  when  it 
is  5  P.M.  at  Chicago  ? 

11.  The  longitude  of   St.   Paul  is   93°  4'  55"  west,  of 
Philadelphia  is  75°  10'  west.     WThat  is  the  difference  in 
longitude  ? 

12.  The  longitude  of  New  York  is  74°  3"  west,  of  Paris 
is  2°  20'  12"  east.     What  is  the  difference  in  longitude  ? 

13.  New  York  City  is  74°  4"  west  from  London.    When 
it  is  noon  at  London,  what  is  the  true  time  at  New  York  ? 

/'  14.  The  longitude  of  Boston  is  71°  4'  west,  and  Chicago 
is  87°  36'  west.  Chicago  is  how  far  due  west  from  Boston, 
if  there  are  51.27  miles  in  one  degree  at  their  latitude  ? 

15.  A  person  travelled  until  his  watch  was  3  hours  too 
fast.     In  what  direction  and  how  far  did  he  go  ? 

16.  What  is  the  difference  in  standard  time   between 
Boston  and  Chicago  ? 

17.  If  a  person  goes  from  New  York  to  San  Francisco, 
will  his  watch  be  too  fast  or  too  slow  ?  and  how  much  ? 

18.  The  difference  in  longitude  between  two   places   is 
17°  54'  55".     What  is  the  difference  in  time  ? 


94  SENIOR   ARITHMETIC. 

19.  The  longitude  of  San  Francisco  is  122°  26'  15"  west, 
and  that  of  Cincinnati  is  84°  26'  west.     When  it  is  9  A.M. 
at  San  Francisco,  what  is  the  tL^e  at  Cincinnati  ? 

20.  The  longitude  of  Boston  is  71°  3'  30"  west,  and  that 
of  Paris  is  2°  20'  12"  east.    When  it  is  30  min.  past  2  P.M. 
at  Paris,  what  is  the  tinie  at  Boston  ? 

21.  Chicago  is  87°  38'  west.     When  it  is  27  min.  36  sec. 
past  11  A.M.  at  Chicago,  it  is  10  min.  past  12  M.  at  Wash- 
ington.    What  is  the  longitude  of  Washington  ? 

22.  St.  Louis  is  90°  15'  15"  west  longitude.     A  gentle- 
man arriving  there  from  Boston,  in  71°  3'  30"  west,  finds 
that  his  watch,  which  was  set  in  Boston,   is  not  right. 
What  change  must  he  make  ? 

23.  Mr.  Jones  started  from  Philadelphia,  and  travelled 
until  his  watch  was  1   hour  30   min.   slow.     How  many 
degrees  did  he  travel  ?  and  in  what  direction  ? 

24.  When  it  is  12  o'clock  noon  at  Chicago,  what  time  is 
it  in  a  place  60°  30'  30"  west  of  Chicago  ? 

25.  Two  men  start  from  the  same  place,  and  travel  in 
the  same  direction,  one  going  3  degrees  and  the  other  5 
degrees  per  day.     They  travel   until   their   difference   in 
time  is  4  hours.     How  many  days  are  they  travelling? 


REVIEW  OF  DENOMINATE  NUMBERS. 

192.  1.   Define  a  simple  number ;  denominate  ;  compound. 

2.  For  what  is  linear  measure  used  ? 

3.  For  what  is  square  measure  used  ? 


4.  For  what  is  cubic  measure  used  ? 

5.  For  what  is  liquid  measure  used  ? 

6.  For  what  is  dry  measure   used  ? 


Give  tables. 


KEVIEW   OF   DENOMINATE   NUMBERS.  95 

7.  For  what  is  Troy  weight  used  ?  table.     Avoirdupois 
weight  ?  table.     Apothecaries'  weight  ?  table. 

8.  How  many  grains  in  a  pound  Troy  ?     Avoirdupois  ? 

9.  How  many  grains  in  an  ounce  Troy  ?    Avoirdupois  ? 

10.  What  is  a  long  ton  ?  and  how  used  ? 

11.  How  many  days  in  a  common  year?  a  leap  year? 
AY  hat  is  the  solar  year  ?     Explain  leap  year.     When  does 
the  civil  day  begin  and  end  ? 

12.  What  is  the  use  of  circular  measure  ?     Define  circle, 
circumference,  diameter,  radius,  arc.     Give  table.     What 
is  the  measure  of  an  angle  ?     What  is  a  degree  ?     A  quad- 
rant ?     How  do  we  find  circumference  ?     How  do  we  find 
diameter  ?     What  is  a  right  angle  ? 

13.  What  is  a  surface  ?  a  square  ?  a  rectangle  ?  a  tri- 
angle ? 

14.  Give  rule  to  find  area  of  a  square ;  of  a  rectangle ; 
of  a  triangle;  of  a  circle. 

15.  Define  solid,  rectangular  solid,  cube,  cylinder.     How 
do  we  find  the  volume  of  rectangular  solid  ?  of  a  cylinder  ? 

16.  W^hat  is  reduction  ?     Reduction  ascending  ?     Reduc- 
tion descending  ? 

17.  Define  a   denominate  fraction.     Give   the  different 
kinds  of  reduction  of  denominate  fractions. 

18.  How   do   we   add   compound    numbers  ?    subtract  ? 
multiply  ?  divide  ? 

19.  Give  the  common  method  of  finding  the  difference 
between  dates.     How  do  we  find  the  exact  difference  ? 

20.  What  is  longitude  ?     How  do  we  find  difference  in 
longitude  between  two  places  on  the  same  side  of  a  prime 
meridian  ?     On  opposite  sides  ? 


96  SENIOR    ARITHMETIC. 

*  21.  How  do  we  find  difference  in  longitude  when  differ- 
ence of  time  is  given  ?  How  find  difference  of  time  when 
difference  in  longitude  is  given  ? 

22.  What  is  standard  time  ?     What   are  the  names  of 
the  four  time  belts  ?     In  passing  west  into  a  time  belt, 
how  does  the  traveller  set  his  watch  ?  in  travelling  east  ? 

23.  How  do  we  find  length  when  area  and  breadth  are 
given  ? 

24.  How  do  we    find    length  when   volume,  thickness, 
and  width  are  given  ? 

25.  What  are  the  dimensions  of  a  rectangular  solid  ? 

26.  Define  cancellation,  even  number,  odd  number,  prime 
number,  composite. 

27.  When  are  numbers  prime  to  each  other  ? 

28.  How  many  cubic  feet  in  a  cord  of  wood  or  stone? 
How  long,  wide,  and  high  is  a  cord  of  wood  ?     What  is  a 
cord  foot  ? 

29.  For  what  is  board  measure  used  ?     What  is  a  board 
foot  ?     Give  rule  for  finding  board  feet. 

30.  How  do  we  find  the  capacity  of  bins  ?  cisterns  ? 

THE  METRIC    SYSTEM. 

LINEAR   MEASURE. 

193.  The  standard  unit  of  Linear  Measure  in  the  Metric 
System  is  the  Meter.     It  is  determined  by  taking  one  ten- 
millionth  part  of  the  distance  from  the  earth's  equator  to 
either  of  its  poles,  measured  on  a  meridian.     It  is  equal  to 
39.37  inches. 

QUESTIONS. 

194.  l.    What  denomination  in  the  English  linear  meas- 
ure is  most  nearly  like  the  meter  ? 

2.    Draw  a  line  one  meter  long. 


THE   METRIC    SYSTEM.  97 

3.  Hold  your  hands  one  meter  apart. 

4.  A  meter  is  about  how  many  feet  long  ? 

5.  How  many  meters  long  is  your  schoolroom  ?     Wide  ? 
High? 

6.  About  how  many  meters  in  a  rod  ? 

HOW  THE  TABLE  IS  MADE. 

195.  Divide  a  meter  into  ten  equal  parts.     One  of  these 
parts  is  a  Decimeter.     Dec  is  a  Latin  stem  meaning  tenth. 
About  how  many  inches  long  is  a  decimeter  ?     Show  with 
your  hands  the  length  of  a  decimeter.     What  part  of  a 
meter  is  a  decimeter  ? 

196.  Divide  a  decimeter  into  ten  equal  parts.     One  of 
these  parts  is  a  Centimeter.     Cent  is  a  Latin  stem  meaning 
hundredth.     What  part  of  an  inch  is  a  centimeter  ?     Show 
its  length.     How  many  centimeters  in  one  meter  ?     What 
part  of  a  meter  is  a  centimeter  ? 

197.  Divide  a  centimeter  into  ten  equal  parts.     One  of 
these  parts  is  a  Millimeter.     Mill  is  a  Latin  stem  meaning 
thousandth.     What  part  of  a  meter  is  a  millimeter  ?    How 
many  millimeters  in  a  meter  ?     What  part  of  an  inch  is  a 
millimeter  ? 

198.  Ten   meters   make    one    Dekameter.      Deka    is    a 
Greek  steam  meaning  ten.     How  many  rods  in   a   deka- 
meter  ?     How  many  feet  ?     How  many  dekameters  long 
is  your  schoolroom  ? 

199.  Ten  dekameters  make  one  Hektometer.     Hekto  is  a 
Greek  stem  meaning  hundred.     How  many  meters  in  one 
hektometer  ?     How  many  feet  long  is  a  hektometer  ? 

200.  Ten  hektometers  make  one  Kilometer.     Kilo  is  a 
Greek  stem  meaning  thousand.     How  many  meters  in  one 
kilometer  ?     How  many  feet  ?     What  part  of  a  mile  ? 


98  SENIOR   ARITHMETIC. 

201.  Ten  kilometers  make  one  Myriameter.     Myria  is  a 
Greek  stem  meaning  ten-thousand.     How  many  meters  in 
one  myrianieter  ?     How  many  feet  ?     How  many  miles  ? 

202.  These  statements  may  be  combined  in  the  follow- 
ing table : 

10  Millimeters  (mm.)  =  1  Centimeter  (cm.)    =      .3937+  in. 

10  Centimeters  =  1  Decimeter  (dm.)     =    3.937+  in. 

10  Decimeters  =  1  Meter  (m.)  =39.37+  in. 

10  Meters  =  1  Dekameter  (Dm.)   =  32.808+  ft. 

10  Dekameters  =  1  Hektometer  (Hm.)  =  19.927+  rd. 

10  Hektometers  =  1  Kilometer  (Km.)     =      .621+  mi. 

10  Kilometers  =  1  Myriameter  (Mm.)  =    6.213+  mi. 


203.  REDUCTION. 

1  Myriameter  = 

10  Kilometers  = 

100  Hektometers  = 

1000  Dekameters  = 
10000  Meters 

100000  Decimeters  = 

1000000  Centimeters  == 
10000000  Millimeters. 


1  Millimeter    = 

.01  Centimeter   = 
o 

.001  Decimeter     = 

.0001  Meter 
.00001  Dekameter   = 
.000001  Hektometer  = 
3  .0000001  Kilometer     = 

.00000001  Myriameter. 


• 

W 

o 


204.  The  following  series  of  numbers  read  from  the  top 
downward  is  reduction  ascending ;  read  from  the  bottom 
upward  is  reduction  descending.  All  metric  numbers  may 
be  reduced  in  this  way. 


75689132.  mm. 
7568913.2  cm. 
756891.32  dm. 
75689.132  m. 
7568.9132  Dm. 
756.89132  Hm. 
75.689132  Km. 
7.5689132  Mm. 


All  these  numbers  might  be  read  thus :       75689132. 


THE   METRIC   SYSTEM.  99 

QUESTIONS. 

205.    1.    How  can  a  metric  number  be  reduced  to  higher 
denominations  ?     To  lower  ? 

2.  Eeduce  12345678  mm.  to  cm. ;  to  dm. ;  to  m.  ;   to 
Dm.  ;  to  Hm. ;  to  Km.  ;  to  Mm. 

3.  Eeduce  9.6538714  Mm.  to  Km. ;  to  Hm. ;  to  Dm. ; 
to  m. ;  to  dm. ;  to  cm.  ;  to  mm. 

4.  Eeduce  7  Mm.  to  lower  denominations. 

5.  Eeduce  7  mm.  to  higher  denominations. 

6.  Eeduce  6307.1  m.  to  Km. ;  to  cm. 

7.  Eeduce  31  meters  to  inches. 

8.  Write  2  Mm.   as  meters ;  7  Km. ;  6  Hm. ;  8  Dm. ; 
5  m.  3  dm. ;  2  cm.  ;  9  mm.     Write  them  all  as  one  number. 

9.  Eeduce  1  Mm.  to  feet. 

10.  Write  7  Mm.  and  6  mm.  in  one  number,  as  meters. 
Eeduce  it  to  higher  denominations ;  to  lower. 

11.  Eeduce  .075  Km.  to  cm. 

12.  Eeduce  8  Dm.  and  6  m.  to  Mm.  ;  to  mm. 

13.  Write  75  Km.  and  62  dm.  in  one  number  as  meters  ; 
as  cm. ;  as  Mm. 

14.  State  the  value  of  each  figure  in  30769.543  M. 

15.  A  ship  sails  100  Mm.  in  one  day.     How  many  miles 
does  it  sail  ? 

16.  Give  the  table  of  Metric  Linear  Measure. 

17.  Name  the  standard  unit. 

18.  How  is  it  determined  ? 

19.  What  is  the  scale  of  the  Metric  system  ? 

20.  Name  in  order  the  Latin  and  Greek  stems  used  in 
the  table. 


100  SENIOR   ARITHMETIC. 

SURFACE   MEASURE. 

206.  The  standard  unit  of   surface  measure  is  the  Are 
(pronounced  like  the  English  air). 

The  Are  is  a  square  whose  side  is  one  dekameter.     It  is 
therefore  a  Square  Dekameter. 

QUESTIONS. 

207.  1.    An  are  is  how  many  meters  long  ?     Wide  ? 

2.  How  many  square  meters  does  the  are  contain  ? 

3.  An  are  is  how  many  inches  long  ?     Feet  ? 

4.  The  are  is  about  how  many  rods  long  ? 

5.  About  how  many  square  rods  does  it  contain  ? 

6.  About  how  many  ares  equal  one  acre  ? 

7.  How  many  ares  does  the  floor  of  your  schoolroom 
contain  ? 

8.  Name  all  the  surfaces  you  can  think  of  that  con- 
tain about  one  are. 

TABLE. 

208.  The  table  of  surface  measure,  like  that  of  linear 
measure,  is  made  by  prefixing  the  Latin  and  Greek  stems 
to  the  standard  unit,  thus  : 

10  Centares  =  1  Deciare,    da. 
10  Declares  =  1  Are,          a. 
10  Ares         =  1  Dekare,    Da. 
10  Dekares  =  1  Hektare,  Ha. 

NOTE.  —  The  denominations  of  the  above  table  are  little  used, 
except  the  are,  the  hektare,  and  the  centare,  which  are  employed 
chiefly  in  measurements  of  land. 

209.  Draw   a   square  whose   side    is    one   meter.     How 
many  square  meters  does  it  contain?     It  is  how   many 


THE   METRIC    SYSTEM.  101 

decimeters  on  a  side  ?  How  many  square  decimeters  does 
it  contain  ?  How  many  square  decimeters  make  one 
square  meter  ? 

210.  Draw  a  square  whose  side  is  one  decimeter.     How 
many   square    decimeters    does    it   contain?     How   many 
centimeters  long  and  wide  is  it  ?     How  many  square  centi- 
meters does  it  contain  ?     How  many  square  centimeters  in 
one  square  decimeter  ?     In  the  same  way  find  how  many 
square  millimeters  in  one  square  decimeter. 

How  many  sq.  Meters  =  1  sq.  Dekameter  ? 

How  many  sq.  Dekameters       =  1  sq.  Hektometer  ? 
How  many  sq.  Hektometers    =  1  sq.  Kilometer  ? 

211.  The  answers  to  the  above  questions  form  the  fol- 
lowing table  of   surface  measure,  which   is   used    for  all 
ordinary  surface  measurements  : 

100  sq.  Millimeters  =  1  sq.  Centimeter,  sq.  cm. 
100  sq.  Centimeters  =  1  sq.  Decimeter,  sq.  dm. 
100  sq.  Decimeters  =  1  sq.  Meter,  sq.  m. 

100  sq.  Meters  =  1  sq.  Dekameter,    sq.  Dm. 

100  sq.  Dekameters    =  1  sq.  Hektometer,  sq.  Hm. 
100  sq.  Hektometers  =  1  sq.  Kilometer,     sq.  Km. 


QUESTIONS. 

212.    1.    Which  denomination  of  this  table  is   like  the 
are? 

2.  Like  the  centare  ? 

3.  Like  the  hectare  ? 

4.  How  far  to  the  right  must  the  decimal  point  be 
moved  to  reduce  sq.  m.  to  sq.  dm.  ? 

5.  How  many  places  to  the  left  must  the  decimal  point 
be  moved  to  reduce  sq.  m.  to  sq.  Dm.  ? 


102  SENIOR   ARITHMETIC. 

6.  To  reduce  sq.  mm.  to  sq.  cm.  ? 

7.  To  reduce  sq.  mm.  to  sq.  dm.  ? 

8.  Reduce  5555  ca.  to  Ha. 

9.  Reduce  3333  Ha  to  ca. 

10.  A  field  134  M.  long  and  7  Dm.  wide,  contains  how 
many  sq.  m.  of  land  ? 

11.  How  many  ares  ? 

12.  How  many  Ha.  ? 

13.  How  many  sq.  Dm.  ? 

14.  How  many  sq.  Hm.  ? 

15.  How  many  sq.  cm.  ? 

16.  How  many  sq.  cm.  in  an  oblong  643  cm.  long  and 
2.5  m.  wide  ? 

17.  How  many  sq.  mm.  ? 

18.  How  many  sq.  Km.  ? 

19.  One  hectare  equals  about  how  many  acres  ? 

VOLUME   MEASURE. 

213.  The   unit   chiefly   used   in   measuring   wood    and 
stone   is    the    Stere  (pronounced  stair),  which   is    a  cube 
whose  edge    is    one   meter.      What    denomination   in  the 
English  volume  measure  is  most  nearly  like  the   stere  ? 
How  many  cubic  meters  does  the   stere  contain  ?     How 
many  decisteres  ?     How  many  centisteres  ?     How   many 
millisteres  ? 

QUESTIONS. 

214.  A  cube  whose  edge  is  one  meter  long  contains  how 
many  cubic  meters  ?     It  is  how  many  dm.  long  ?     Wide  ? 
High  ?     How  many  cu.  dm.  does  it  contain  ?     How  many 
cu.  dm.  =  1  cu.  m.  ?     A  cube  whose  edge  is  1  dm.  contains 
how  many  cu.  dm.  ?     How  many  cm.  long  is  it?     Wide? 


THE   METRIC    SYSTEM.  103 

High  ?  How  many  cu.  cm.  does  it  contain  ?  How  many 
cu.  cm.  =  1  cu.  din.  ?  A  cube  whose  edge  is  1  CHI.  con- 
tains how  many  cu.  cm.  ?  How  many  mm.  long  is  it  ? 
Wide  ?  High  ?  How  many  cu.  mm.  does  it  contain  ? 
How  many  cu.  mm.  =  1  cu.  cm.  ? 

215.    From  the  answers  to  the  above  questions  make  the 


following  : 


TABLE  OF  VOLUME  MEASURE. 


1000  cu.  Millimeters  =  1  cu.  Centimeter,  cu.  dm. 
1000  cu.  Centimeters  =  1  cu.  Decimeter,    cu.  m. 
1000  cu.  Decimeters  =  1  cu.  Meter,  cu.  cm. 

QUESTIONS. 

216.  1.    How  may  cubic  millimeters  be  reduced  to  cubic 
centimeters  ?     To  cubic  dm.  ?     To  cu.  m.  ? 

2.  How  many  places  to  the  right   must  the  decimal 
point  be  moved  to  reduce  cu.  meters  to  cu.  millimeters  ? 

3.  Reduce  7  cu.  meters  to  cu.  millimeters. 

4.  Reduce  5  cu.  millimeters  to  cu.  meters. 

5.  How  many  steres  in  one  cu.  meter  ? 

6.  A  pile  of  wood  is  30  dm.  long,  3  m.  wide,  and  18  dm. 
high.     How  many  cu.  meters  does  it  contain  ? 

7.  How  many  steres  ? 

8.  How  many  cu.  millimeters  ? 

9.  How  many  cu.  centimeters  of  air  in  an  empty  box 
2  m.  by  12  dm.  by  75  cm.  ? 

10.  How  many  cubic  dm.  ? 

11.  How  many  steres  of    stone   in  a  wall  30  m.  long, 
5  dm.  thick,  and  250  cm.  high  ? 


104  SENIOR    ARITHMETIC. 

CAPACITY  MEASURE. 

217.  The  metric  capacity  measure  takes  the  place  of  both 
the  liquid  and  the  dry  measure  of  the  English  system. 

The  standard  unit  of  capacity  measure  is  the  Liter 
(pronounced  teeter),  which  is  a  cube  whose  edge  is  one 
decimeter. 

QUESTIONS. 

218.  1.  The  liter  is  what  part  of  a  meter  wide  ?     High  ? 
Long  ? 

2.    What  part  of  a  cubic  meter  does  it  contain  ? 
'3.    About  how  many  inches  wide  is  it  ?    High  ?    Long  ? 
About  how  many  cu.  in.  does  it  contain  ? 

4.  Show  with  your  hands  how  wide,  high,  and  long  a 
liter  is. 

5.  What  denomination  of  English  dry  measure  corre- 
sponds most  nearly  to  the  liter  ? 

6.  Make  a  full-sized  picture  of  a  liter. 

7.  What  object  the  size  of  a  liter  do  you  know  ? 

TABLE. 

219.  The  table  of  capacity  is  formed  similarly  to  the 
other  metric  tables,  and  is  as  follows  :  — 

10  Milliliters  (ml.)  =  1  Centiliter,  c. 

10  Centiliters  =  1  Deciliter,  dl. 

10  Deciliters  =  1  Liter,  1. 

10  Liters  =  1  Dekaliter,  Dl. 

10  Dekaliters  =  1  Hectoliter.  HI. 

10  Hektoliters  =  1  Kiloliter,  Kl. 

10  Kiloliters  =  1  Myrialiter,  Ml. 


QUESTIONS. 

220.  1.    How  many  liters  in  1  myrialiter  ?     In  1  ml.  ? 
2.    How  many  milliliters  in  1 


ML? 


THE   METRIC    SYSTEM.  105 

3.  Reduce  12345678  ml.  to  higher  denominations. 

4.  Eead  the  above  number,  giving  each  figure  the  name 
of  the  denomination  it  represents. 

5.  Eeduce  154.67  cl.  to  Kl. 

6.  Reduce  .012346  Ml.  to  dl. 

7.  How  many  liters  equal  one  cubic  meter  ? 

8.  A  bin  is  2.5  in.  wide,  6.4  in.  long,  and  17  dm.  deep. 
How  many  liters  of  oats  will  it  hold  ?     How  many  HI.  ? 
How  many  Kl.  ? 

9.  A  tank  is  3  m.  long,  and  3  m.  wide.     How  many 
dm.  deep  must  it  be  to  hold  50  HI.  of  water  ? 

10.    A  stone  containing  1  stere,  if  dropped  in  a  pond, 
would  displace  how  many  liters  of  water  ? 

MEASURES   OF  WEIGHT. 

221.  The  Gram  is  the  unit  of  weight.     It  is  equal  to 
the  weight  of  a  cubic  centimeter  of  distilled  water  at  its 
greatest  density. 

TABLE. 

10  Milligrams  (mg.)  =  1  Centigram,  eg. 

10  Centigrams  =  1  Decigram,  dg. 

10  Decigrams  =  1  Gram,  g. 

10  Grams  =  1  Dekagram,  Dg. 

10  Dekagrams  =  1  Hektogram,  Hg. 

10  Hektograms  =  1  Kilogram,  Kg. 

10  Kilograms  =  1  Myriagram,  Mg 

10  Myriagrams  =  1  Quintal,  Q. 

10  Quintals  =  1  Tonneau,  T. 

or  Metric  Ton. 

QUESTIONS. 

222.  1.    How  many  grams  in  1  Metric  Ton  ? 
2.    How  many  mg.  in  1  metric  ton  ? 


106  SENIOR   ARITHMETIC. 

3.  Keduce  1  mg.  to  T. 

4.  Reduce  1  T.  to  mg. 

5.  Keduce  9876543215  mg.  to  higher  denominations. 

6.  Read  the  above  number,  giving  each  figure  the  name 
of  the  denomination  it  represents. 

7.  Recite  the  table  of  weight. 

8.  Spell  the  name  of  each  denomination. 

9.  Reduce  7.42  quintals  to  centigrams. 

10.  Reduce  543  mg.  to  Mg. 

11.  One  gram  equals  15.432  grains.     How  many  grains 
in  1  Kg.  ? 

12.  One   pound,   Avoirdupois,   contains  7000  gr.     How 
many  pounds  are  equivalent  to  one  Kg.  ? 

13.  Mr.  Smith  weighs  100  Kg. ;  how  many  pounds  does 
he  weigh  ? 

14.  How  many  grams  does   a  cubic  meter  of   distilled 
water  weigh  ? 

15.  Would  a  cubic  meter  of  any  other  substance  weigh 
the  same  ?     State  your  reason. 

16.  How  many  kilograms  of  water  will  a  tank  4  m.  X 
3  m.  X  12  dm.  hold  ? 

REVIEW   QUESTIONS. 

223.  l.    How  many  tables  in  the  Metric  System  ? 

2.  Name  the  standard  units  in  the  order  in  which  they 
have  been  given.     Repeat  them  until  you  can  say  them  as 
rapidly  as  you  can  talk. 

3.  Name  the  prefixes  in  the  same  way. 

4.  Name  and  describe   the  unit  of  capacity  measure ; 
of  weight ;  of  length ;  of  volume ;  of  surface. 

5.  Repeat  the  tables. 


GENERAL   REVIEW.  107 

6.  The  stere  is  the  unit  of  what  measure  ?    The  meter  ? 
The  are  ?     The  gram  ?     The  liter  ? 

7.  How  can  metric  numbers  be  reduced  to  higher  de- 
nominations ?  to  lower  ? 

8.  How  many  things  are  to  be  committed  to  memory 
in  the  Metric  System  ? 

9.  What  is  39.37  ?  15.432  ?  10  ?    These  are  the  only 
numbers  that  need  be  remembered. 


GENERAL   REVIEW, 

224.  l.    Define  fraction,  numerator,  fractional  unit,  terms, 
reduction  of  fractions. 

2.  Change  217  to  20ths. 

3.  Give  the  principle  upon  which  reduction  of  fractions 
is  based.     Illustrate. 

^x4.    Add  25i,  14§,  7f. 

5.  Give  the  rule  for  reducing  fractions  to  their  least 
common  denominator. 

6.  A  man  owned  §  of  a  foundry  and  sold  \  of  his  share 
for  $1200  ;  what  was  the  foundry  worth  ? 

7.  Reduce  to  simple  form  (15-|  —  3|)  X  (2£  +  5|). 


8 


9.  Reduce  to  least  common  denominator  six  thirty- 
fifths,  nine  twentieths,  and  five  sixteenths,  and  arrange  the 
results  according  to  value. 

10.    A  man  having  $130  used  f  of  it  ;  how  much  of  it 
remained  ? 

1.  C  and  D  can  do  a  piece  of  work  in  24  days,  D  can 
do  it  alone  in  45  days  ;  how  many  days  will  C  require  to 
do  it? 


108  SENIOR    ARITHMETIC. 

12.  The  numerator  of  a  fraction  is  6510,  the  denomi- 
nator 66495;  reduce  the  fraction  to  its  lowest  terms. 

13.  If  7  be  added  to  each  term  of  the  fraction  |,  will  its 
value  be  increased  or  diminished,  and  how  much  ? 

14.  Two  men   are  140  miles  apart,  and  travel  towards 
each  other,  one  at  the  rate  of  3|  miles  an  hour,  and  the 
other  at  the  rate  of  41  miles  an  hour;  in  how  many  hours 
will  they  meet  ? 

15.  Define  decimal  fraction  ;  an  account ;  currency. 

16.  What   will   6827    feet   of    lumber   cost    at    $10.50 
per-  M.  ? 

17.  8.7625  +  31.735  -  17.382569  =  ? 

18.  Write  in  words  365.     8752. 

19.  Find  the  cost  of  7896  pounds  of  hay  at  $16  a  ton. 

20.  Express  in  figures  two  hundred  sixty-five  and  five 
thousand  one  hundred  ten  millionths. 

21.  Change  .875  to   a  common   fraction   in   its    lowest 
terms. 

22.  How  is  a  bill  receipted  ? 

23.  Give  a  rule  for  dividing  a  decimal  by  10,  100,  1000, 
etc. 

24.  Keduce  3.25,  12.364,  and   .56087  to  a  common  de- 
nominator. 

25.  When  will  a  fraction  reduce  to  a  perfect  decimal  ? 

26.  7.6875  -f-  187.5  x  (5|  +  2|)  =  what  ? 

27.  How  is  the  place  for  the  decimal  point  in  the  pro- 
duct determined  ? 

28.  Give  the  abbreviations  of  Creditor  and  Merchandise. 

29.  Name  the  seventh  decimal  order. 

30.  James   Harris  of  Syracuse,  N.Y.,  sold  for  cash  to 
Preston  White,  on  Nov.  4,  1887,  42   Ib.   of    sugar  at  10 


GENERAL    REVIEW.  109 

cents ;  3  Ib.  Y.  H.  tea  at  $.60 ;  4  gal.  molasses  at  $.75 ; 
48  yd.  sheeting  at  $.14 ;  1  box  starch  46  cents  j  and  8  doz. 
eggs  at  $.24. 

Make  the  bill  in  due  form. 

31.  Define  a  square  ;  a  circle. 

32.  Write  the  table  of  cubic  measure. 

33.  For  what  purposes  are    the    following   used :  Troy 
weight,  dry  measure  ? 

34.  The   last  war  with  England  commenced  June    18, 
1812,  and  ended  Feb.  17,  1815.    How  long  did  it  continue  ? 

35.  A  jeweller  made  3  Ib.  2  pwt.  2  gr.  of  gold  into  rings 
weighing   5    pwt.    10    gr.    each.     How   many   rings   were 
there  ? 

36.  Reduce  2  mi.  6  ch.  3  rd.  to  links. 

37.  Reduce  to  integers  of  lower  denominations  £|,  and 
.25256  T. 

38.  Change  4  %  5  3  2  9  8  gr.  to  a  decimal  of  a  pound. 

39.  Find  the  result  of  4.8  bu.  -f  2f  bu.  -f  .8125  pk.  + 
2|  pk.  +  |  bu. 

40.  A  grocer  bought  35  casks  of  molasses,  eaoh  contain- 
ing 44  gal.  2  qt.  1  pt.     How  much  did  they  all  contain  ? 

41.  A  ship  in  8°  north  latitude  sailed  due  south  until  it 
reached  12°  south  latitude ;  find  the  distance  it  sailed  in 
statute  miles. 

42.  Find  the  value  of  T75  of  a  ton. 

43.  Reduce  ^  of  a  year  to  integers  of  lower  denomi- 
nations. 

44.  Reduce  f  of  a  Ib.  Troy  to  integers  of  lower  denomi- 
nations. 

45.  Express  120  rd.  2  yd.  1  ft.  6  in.  as  the  fraction  of 
a  mile. 


110  SENIOR    ARITHMETIC. 

46.  Eeduce  45  sq.  rd.  2  sq.  ft.  9  sq.  in.  to  the  fraction  of 
an  acre. 

47.  ^hat  part  of  a  day  are  6  hr.  13  min.  20  sec.  ? 

48.  What  part  of  4  gal.  2  qt.   1  pt.  are   1   gal.  1   qt. 
1  pt.  ? 

49.  At  25  cents  an  ounce,  what  is  the  value  of  18  oz.  10 
pwt.  12  gr.  of  silver  ? 

50.  How  much  will  it  cost  to  fill  a  bin  with  corn  at  $.45 
a  bushel,  if  the  bin  is  10  ft.  square  on  the  bottom  and  4 
ft.  deep  ? 

51.  A  cistern  measures  inside  the  walls  8  by  6  by  9  ft., 
and  lacks  f^  ft.  of  being  full.     How  many  gallons  does  it 
hold  ? 

52.  How  many  bushels  will  a  box  of  the  same  dimen- 
sions hold  ? 

53.  How  many  cords  of  wood  in  a  pile  of  wood  that  is 
twice  the  length,  height,  and  width  of  an  established  cord  ? 

54.  Find  the  total  weight  of  5  car-loads  of  coal,  weigh- 
ing respectively  14  T.  18  cwt.  63  lb.,  17  T.  4  cwt.  85  lb., 
13  T.  19  cwt.  26  lb.,  15  T.  10  cwt.  43  lb.,  and  14  T.  7  cwt. 
90  lb. 

55.  How  many  bricks  8  in.  by  4  in.  by  2  in.  will  it  take 
to  pave  a  street  ^  mile  long  and  ^  mile  wide,  laying  the 
brick  011  the  longest  narrow  face  ?     How  many  if  they  are 
placed  on  end  ? 

56.  How  much  wood  in  three  piles,  the  first  of  which 
contains  10  cd.  6  cd.  ft.  4  cu.  ft,  the  second  12  cd.  12  cu. 
ft.,  the  third  17  cd.  1  cd.  ft.  ? 

57.  A  family  consumes  daily  6  lb.  14  oz.  of  bread.     If 
each  loaf  weighs  1  lb.  6  oz.  and  costs  7  cents,  how  much 
does  bread  cost  the  family  for  the  month  of  August  ? 

58.  Find  the  sum  of  g  mi.,  §  fur.,  £  rd.,  and  §  ft. 


GENERAL   REVIEW.  Ill 

59.  From  6J  mi.  take  4  mi.  140  rd.  4  yd. 

60.  A  man  has  a  bin  6  ft.  long,  4  ft.  wide,  3  ft.  deep, 
§  filled  with  wheat.     If  he  sells  10  sacks,  each  containing 
2  bu.  1  pk.  5  qt.,  how  much  is  left  ? 

61.  From  a  piece  of  land  20  rods  long,  180  ft.  wide,  were 
sold  4  lots,  each  50  ft.  wide,  150  ft.  long.     What   part 
remained  ? 

62.  A  merchant  bought  two  casks  of  wine,  each  contain- 
ing 41  gal.  3  qt.,  at  $1.80  per  gallon.     One-seventh  of  it 
leaked  away.     He  sold  9  kegs,  each  containing  5  gal  1  qt. 
at  30/  a  pint,  and  the  remainder  for  40/  a  pint.     How 
much  did  he  gain  ? 

63.  A  coal-dealer  bought  34160  Ib.  of  coal  at  $2.50  per 
long  ton.     He  sold  8  loads,  each  1  T.  4  cwt.  60  Ib.,  at  $3 
per  short  ton,  and  the  rest  for  $3.25  per  short  ton.     How 
much  did  he  gain  ? 

64.  Change  3,895,504"  to  higher  denominations. 

65.  Three  quadrants  of  a  circle  are  equal  to  how  many 
seconds  ? 

66.  Through  how  many  degrees  does  the  minute-hand 
of  a  clock  pass  in  1\  hours  ?    Through  how  many  does  the 
hour-hand  pass  in  the  same  time  ? 

67.  How  many  minutes  elapse  between  four  o'clock  Fri- 
day  afternoon   and   nine   o'clock    the   following   Monday 
morning  ? 

68.  A  boy  was  exactly  10  years  old  when  the  United 
States  declared  war  against  Mexico,  May  13,  1846.     How 
old  was  he  at  the  time  of  the  first  bloodshed  of  the  Civil 
War,  April  19,  1861  ? 

69.  A  train  leaves  New  York  at   six  o'clock  Monday 
evening,  and  travels  an  average  of  §  of  a  mile  a  minute. 
When  will  it  reach  Buffalo,  a  distance  of  410  miles  ? 


112  SENIOR   ARITHMETIC. 

70.  A  cistern  that  holds  50  bushels  is  6  ft.  square ;  how 
deep  is  it  ? 

71.  A  pile  of  wood  is  6  ft.  high  and  4  ft.  wide.  -  How 
long  must  it  be  to  contain  3  cords  ? 

72.  A  man  has   a  circular  garden  with  a  diameter  of 
36  feet.     How  many  rods  of  fencing  will  be  required  to 
enclose  it  ? 

73.  How  many  square  yards  in  the  above  garden  ? 

74.  A  city  lot  is  35  ft.  front  and  125  ft.   deep.     Find 
the  area. 

75.  A  meadow  contains  8|  acres.     Its  width  is  35  rods  ; 
find  the  length  of  it. 

76.  How  many  yards  of  carpeting  f  of  a  yd.  wide  will 
be  required  for  a  room  18  ft.  wide  and  20  ft.  long,  if  the 
strips  run  lengthwise,  and  there  is  a  waste  of  6  in.  in  each 
strip  for  matching  patterns  ? 

77.  The  platform  in  a  schoolroom  is  30  ft.  long  and  11 
ft.  wide.     What  will  be  the  cost  of  oil-cloth,  at  85  cents 
a  sq.  yd.,  to  cover  it  ? 

78.  How  many  feet,  board  measure,  in  6  boards  16  ft. 
long,  10  in.  wide,  1  in.  thick  ? 

79.  Find  the  cost  of  10  Norway  sidewalk  planks  16  ft. 
long,  12  in.  wide,  2  in.  thick,"  at  $18  per  M. 

80.  A  class-room  is  15  ft.  long,  12  ft.  wide,  10  ft.  high. 
Find  the  cost  of  plastering  it  at  20  cents  a  yard. 

81.  A  room  is  30  ft.  long,  40  ft.  wide,  and  16  ft.  high. 
Find  the  number  of  square  yards  of  plastering  in  it  after 
making  allowance  for  wainscoting  3  ft.  high,  8  windows, 
4  ft.  by  8  ft.,  and  6  doors,  3  ft.  6  in.  by  7  ft.  6  in. 

82.  What  will  it  cost  to  paper  a  kitchen  12  ft.  by  11  ft. 
and  9  ft.  high,  with  10-cent  paper,  if  each  roll  covers  4  sq. 

yd.? 


GENERAL    REVIEW.  113 

X83.  Find  the  cost  of  papering  a  room  16  ft.  long,  12  ft. 
wide,  9  ft.  6  in.  high,  with  paper  18  in.  wide,  8  yards  in  a 
roll,  at  50  cents  a  roll,  if  20  sq.  yd.  be  allowed  for  doors, 
windows,  and  base-boards. 

84.  If  a  shingle  is  4  inches  wide,  and  lays  5i-  inches  to 
the  weather,  how  many  shingles  will  it  take  to  shingle  one 
side  of  a  roof  that  is  32  ft.  long  by  22  ft.  wide,  allowing  ail 
extra  course  at  the  eaves  ?     How  many  for  both  sides  ? 

85.  What  would  be  the  cost  of  the  shingles  for  both  sides 
of  the  roof  in  No.  84  at  $3.25  per  M.  ? 

86.  The  product  of  two  numbers  is  l^y  ;  one  of  the  num- 
bers is  ^.     What  is  the  other? 

87.  What  fraction  multiplied  by  f  will  equal  T3T  ? 

88.  How  many  square  yards  of  carpet  will  be  required 
to  carpet  a  room  that  is  27  ft.  by  33  ft.  ?     How  many  yards 
of  carpet  will  be  required  if  the  carpet  is  30  inches  wide  ? 

89.  If  a  hotel  uses  3  pounds  of   coifee  a  week,  what 
would  be  paid  for  coffee  at  38  cents  a  pound  for  January, 
February,  and  March,  1896  ? 

90.  When  it  is  noon  at  New  York,  73°  59'  9"  W.,  what 
is  the  time  at  Chicago,  87°  36'  42"  W.  ?     What  is  the  time 
at  New  York  when  it  is  noon  at  Chicago  ? 

91.  When  it  is  noon  at  Greenwich,  what  is  the  longi- 
tude of  a  place  whose  time  is  8.30  A.M. 

s  92.  A  and  B  start  at  a  given  point,  and  travel  in  opposite 
directions.  A  travels  until  his  longitude  is  30°  40'  greater 
than  it  was,  and  B  travels  half  as  far  as  A.  What  is  the 
difference  in  time  between  the  places  they  are  then  in  ? 

93.  What   part   of  a   pound    Avoirdupois   is    a    pound 
Troy? 

94.  What  part  of  an  ounce  Troy  is  an  ounce  Avoirdu- 
pois ? 


114  SENIOR   ARITHMETIC. 

95.  A  druggist  bought  opium  at  $8  a  pound  Avoirdupois, 
and  sold  it  at  75/  an  ounce  Troy.     What  was  his  profit  on 

10  pounds  ? 

96.  How  much  heavier  is  a  pound  of  iron  than  a  pound 
of  gold  ? 

97.  What  is  the  difference  in  the  areas  of  two  fields, 
one  being  5  Hm.  long  and  8  Dm.  wide,  the  other  8  Hm. 
long  and  14  Dm.  wide  ? 

98.  In  a  cubic  dekameter  how  many  cubic  millimeters  ? 

99.  A  rectangular  field  is  5.4  Hm.  long  and  1.5  Hm. 
wide.     How  many  hektares  does  it  contain  ? 

100.  Three  fields  have  an  area  respectively  of  19  A.  146 
sq.  rd.,  12  A.  73  sq.  rd.  15  sq.  yd.,  and  9  A.  127  sq.  rd.  26 
sq.  yd.     What  is  the  total  area  ? 

101.  Find  the  volume  and  the  area  of  the  curved  surface 
of  a  cylinder  whose  diameter  is  8  in.,  and  whose  altitude  is 

11  in. 

PERCENTAGE. 
225.    Oral. 

How  much  is  1  of  20  ? 
5  is  i  of  what  ? 

5  is  what  part  of  20  ?     5  is  how  many  hundredths  of  20  ? 
Questions  of  Relation  may  be  solved  by  means  of  hun- 
dredths ;  thus, 

a.  How  much  is  T2^  of  20  ? 

b.  5  is  T2<£y  of  what  ? 

c.  5  is  what  part  of  20  ?     5  is  how  many  hundredths  of 
20? 

Another  name  for  hundredths  is  per  cent ;  thus,  y2^  is 
25  per  cent,  T§&  =  8  per  cent,  .16  =  16  per  cent,  .05  = 
5  per  cent. 


PERCENTAGE.  115 

The  sign  of  per  cent  is  %.  25  per  cent  is  25%,  6  per 
cent  is  6%,  £  per  cent  is  J-%. 

Read  questions  a,  b,  and  c,  using  the  name  per  cent 
where  necessary. 

Write  questions  a,  b,  and  c,  using  the  sign  %  in  its 
proper  place. 

Solve  questions  a,  b,  and  c,  using  decimal  per  cent. 

226.  Percentage  is  a  process  of  solving  questions  of  rela- 
tion by  means  of  hundredths. 

Written. 

1.  How  much  is  3%  of  400  ? 

Solve  the  above,  then  form  question  b,  and  solve  it. 

2.  How  much  is  10%  of  200  ? 

Solve  the  above,  form  question  c,  and  solve  it. 
Solve  the  following  questions,  then  form  questions  b  and 
c,  and  solve  them. 

3.  How  much  is  4  per  cent  of  $200  ? 

4.  30%  of  500  is  how  much  ? 

5.  How  much  is  50%  of  90  ? 

6.  A  boy  earned  §4.00,  and  spent  10%  of  it  for  a  book. 
What  was  the  cost  of  the  book  ?     (Question  a.) 

7.  A  farmer  had  100  sheep  and  sold  20%  of  them. 
How  many  sheep  did  he  sell  ? 

Solve  the  following,  then  form  questions  a  and  c,  and 
solve  them. 

^—-8.    15  is  10%  of  what  number  ? 
*  9.    160  is  80%  of  what  number  ? 

10.  50  is  25  per  cent  of  what  number  ? 

11.  A  boy  lost  20  cents,  which  was  5  per  cent  of  all  his 
money.     How  much  money  did  he  have  ?     (Question  b.) 


116  SENIOR   ARITHMETIC. 

12.  A  farmer  sold  150  sheep,  which  was  50%  of  his 
entire  flock.     How  many  sheep  were  in  the  flock  ? 

Solve  the  following,  then  form  questions  a  and  b,  and 
solve  them. 

13.  $20  is  what  per  cent  of  $100  ? 

14.  What  per  cent  of  60  is  15  ? 

15.  40/  is  what  %  of  $4.00  ? 

16.  A  boy  earns  $5.00  a  week,  and  saves  $2.00  of  it. 
What  per  cent  of  his  money  does  he  save  ?     (Question  c  ) 

17.  A  dealer  bought  a  gross  of  pencils,  and  sold  36  of 
them.     What  per  cent  of  his  pencils  did  he  sell  ?     What 
per  cent  remained  unsold  ? 

Change  the  following  fractions  to  others  having  100  foi* 
a  denominator  :  J  ;  .»  ;  ft  ;  J  ;  ft  ;  ft  ;  i  ;  j  ;  ft  ;  ft- 

Change  the  above  fractions  to  decimal  hundredths. 

Read  as  hundredths:  .05;  .186;  .33£;  .24£;  .27|;  .2725; 
.5  ;  .1. 

Write  the  above  in  hundredths  as  common  fractions. 

227.  Read  the  following  Questions  of  Relation. 
Question  a.   How  much  is  5%  of  200  ?     Ans.  10. 
Question  b.    10  is  5%  of  what  ?     Ans.  200. 
Question  c.    10  is  what  %  of  200  ?     Ans.  5%. 

These  three  kinds  of  questions  form  the  basis  of  a  great 
variety  of  practical  computations,  which  are  classed  under 
the  general  head  of  Percentage. 

228.  Every  question  in  percentage  involves  three  ele- 
ments :  the  Rate  per  cent,  the  Base,  and  the  Percentage. 

The  Rate  per  cent  is  the  number  of  hundredths  taken. 
In  question  a,  what  is  the  rate  per  cent  ? 

The  Base  is  the  number  of  which  the  hundredths  are 
taken.  In  question  a,  what  is  the  base  ? 


PERCENTAGE.  117 

The  Percentage  is  the  result  obtained  by  taking  a  cer- 
tain per  cent  of  a  number.  In  question  a,  what  is  the 
percentage  ? 

How  much  is  8%  of  $200  ? 

SOLUTION.  —8%  of  $200  =  200  x  .08  =  $16.  We  now  have  the 
three  elements,  as  follows : 

8/0  is  the  rate,  $200  is  the  base,  and  $16  is  the  percentage. 
Since  $200  x  .08  =  $16,  the  percentage  ; 
$16  -f-  .08  =  8200,  the  base  ; 
And  $16  -j-  $200  =  .08,  the  rate. 

229.  Therefore,  when  any  two  of   these  elements  are 
given,  the  other  may  be  found,  thus  : 

Base  X  Rate  =  Percentage ; 
Percentage  -f-  Rate  =  Base ; 
Percentage  -v-  Base  =  Kate. 

230.  Tell  which  elements  are  given,  and  which  one  is 
required,  in  question  a ;  in  question  b  ;  in  question  c. 

231.  Find  the  percentage  and  form  questions  b  and  c, 
but  do  not  solve  them. 

18.  6%  of  100  is  what? 

19.  How  much  is  25%  of  200  ? 

20.  How  much  is  40%  of  250  ? 

21.  What  is  4%  of  50  men  ? 

22.  20%  of  80  is  what  ? 

23.  15%  of  $40  =  ? 

24.  What  is  3%  of  400  gallons  ? 

25.  What  is  90%  of  200  pounds  ? 

26.  60%  of  200  miles  =  ? 

27.  10%  of  15  inches  =  ? 

28.  What  is  the  base  in  each  of  the  above  questions  ? 


118  SENIOR   ARITHMETIC. 

232.  Care  should  be  taken  to  express  the  decimal  rate 
per  cent  properly,  as  hundredths.     Every  fractional  part 
of  1%  must  be  written  at  the  right  of  the  hundredths 
place. 

1%  =  .01.  12i%  =  .121  or  .125. 

9%  =  .09.  i%  =  .001  or  .005. 

10%  =  .10.  10T^%  =  -10^ 

90%  =  .90.  33^%  =  .33  J. 

100%  =  1.00.  81%  =  .081  or  .0825. 

900%  =  9.00.  |%  =  .OOi  or  .0025. 

125%  =  1.25.  \%  =  .00£  or  .00125. 

233.  Express  decimally : 


1.    7% 

6.    6i% 

11.    101% 

16.    i% 

2.    6% 

7.    12i% 

12.    110% 

17.    |% 

3.    2% 

8.    15|% 

13.    250% 

18-    f% 

4.    12% 

9.    37£% 

14.    200% 

19.    |% 

5.    78% 

10.    4|% 

15.    127i% 

20.    ^ 

234.    It  is  often  convenient  to  change  the  rate  per  cent 
to  the  common  fraction  form  ;  thus  : 

L2i  =  J£  =  #  X  -±-    -i 

100    100    2       ~ 


4 

Change  to  common  fractions  in  lowest  terms  : 

1.  25%  5.  16|%  9.  150%  13.  f% 

2.  50%  6.  33i%  10.  225%  14.  f% 

3.  75%  7.  37i%  11.  175%  15.  f% 

4.  20%  8.  87i%  12.  236%  16.  1% 

What  per  cent  of  a  number  isjofit?£?§?j?j? 

M^ft*  A?  i?  I?  *?  I*  ft*  /5?  *?'  A?  iV? 

?    T90?    i?    1? 


PERCENTAGE.  119 

235.  Express   in   both   the   decimal   and   the    common 
fraction  form  : 

1.  25%  5.  6|%  9.  108%  13.  f% 

2.  60%  6.  6J%  10.  150%  14.  f% 

3.  18%  7.  1\%  11.  125%  15.  1% 

4.  1%  8.  66|%  12.  1371%  16.  T7o% 

236.  Per  cent  is  commonly  used  in  the  decimal  form, 
but  many  operations  may  be  much  shortened  by  using  the 
common  fraction  form. 

Solve,  using  first  the  decimal,  then  the  common  fraction 
form,  and  note  the  difference. 

17.  How  much  is  25%  of  $324  ? 

18.  Find  12 J%  of  960  sheep. 

19.  What  is  16f  %  of  366  men? 

20.  Find  33  J%  of  12  oranges. 

21.  50%  of  4  tons  is  what  ? 

22.  20%  of  $300  =  what  ? 

Question  a,  Oral. 

23.  What  is  ^  of  800  ?  27.  50%  of  144  men  ? 

24.  What  is  Tfo  of  900  ?  28.  20%  of  15  eggs  ? 

25.  16f  %  of  48  apples  ?  29.  .07  of  500  ? 

26.  33J%  of  66  sheep  ?  30.  12|-%  of  $16  ? 

Written. 

237.  Rate  and  base  given,  to  find  percentage. 
Base  X  Bate  =  Percentage. 

1.  What  is  40%  of  $120  ? 

2.  How  much  is  12  J%  of  1600  Ib.  ? 

3.  18T\  of  365  is  what  ? 

4.  From  a  flock  of  60  sheep,  10%  were   sold.     How 
many  were  sold  ?     What  is  the  question  in  this  problem  ? 


120  SENIOR    ARITHMETIC. 

5.  How  much  is  100%  of  50  bushels  ? 

6.  A  man  having  50  bushels  of  wheat  sold  20' per  cent 
of  it.     How  many  bushels  did  he  sell  ? 

7.  A  man  had  $1500  in  the  bank  and  drew  out  40% 
of  it.     How  much  remained  in  the  bank  ? 

NOTE.  —  100 %  represents  all  he  had  in  the  bank.  100^  —  40% 
=  60%,  the  part  that  remained.  The  question  then  becomes,  How 
much  is  60%  of  $1500  ? 


8.  A  farmer  having  320  acres  of  land  sold  15%  of  it 
to   one  man  and  25%  to  another.     How  many  acres  did 
he  sell  ? 

9.  A  wholesale  grocer  had  480  bbl.  of  A  sugar,  and  sold 
12^%  of  it.     How  much  remained  unsold? 

10.  How  much  is  .5%  of  80  ? 

11.  How  much  is  J%  of  $4000  ? 

Question  t>,  Oral. 

1.  5  is  ^V  of  what  ?  4.  12  is  8%  of  what  ? 

2.  5  is  25%  of  what?  5.  30  is  12i%  of  what? 

3.  40  is  10%  of  what  ?  6.  15  is  50%  of  what .' 

Written. 

238.    Percentage  and  rate  given,  to  find  base. 

Percentage  -5-  Eate  =  Base. 

7.  $125  is  12i%  of  what? 

8.  150  bu.  is  33i%  of  what? 

9.  240  is  120%  of  what? 

10.  $1644  is  40%  of  what? 

11.  75  is  3i%  of  what? 

12.  289  is  50%  of  what? 

13.  25%  of  my  property  is  $5000.     What  is  the  value 
of  my  property  ? 


PERCENTAGE.  121 

14.    I  sold  a  horse  for  $81,  which  was  90%  of  what  it 
cost  me.     What  did  the  horse  cost  me  ? 

Question  <•,  Oral. 

1.  What  part  of  45  is  15  ? 

2.  What  per  cent  of  45  is  15  ? 

3.  What  per  cent  of  80  is  60  ? 

4.  AVhat  per  cent  of  90  is  30  ? 

5.  $40  is  what  per  cent  of  $60  ? 

6.  12  yd.  is  what  per  cent  of  36  yd.  ? 

7.  14  bu.  is  what  per  cent  of  56  bu.  '/ 

8.  f  is  what  per  cent  of  *  ? 

Written. 

239.  Base  and  percentage  given,  to  find  rate. 

Percentage  -j-  Base  =  Kate. 

9.  What  per  cent  of  $240  is  $80  ? 

10.  150  is  what  per  cent  of  900  ? 

11.  What  %  of  a  long  ton  is  a  short  ton  ? 

12.  What  %  of  5  days  is  6  hours  ? 

13.  5  cwt.  is  what  %  of  3  tons  ? 

14.  $28.16  is  what  %  of  $7040  ? 

15.  What  per  cent  is  J  of  2£  ?     £  of  §  ?     f  of  7£  ? 

16.  My  salary  is  $1600  and  my  expenses  $1200.     What 
%  of  my  salary  are  my  expenses  ? 

240.  The  sum  of  the  base  and  percentage  is  called  the 
Amount. 

241.  The  difference  between  the  base  and  percentage  is 
called  the  Difference. 

1.    Find  the  amount  in  the  following : 
How  much  is  10%  of  20  ?     Find  the  difference. 


122  SENIOR   ARITHMETIC. 

2.  20  is  what  per  cent  of  itself  ? 

3.  If  20  is  increased  by  10%  of  itself,  the  amount  is 
22.     What  per  cent  of  20  is  22  ? 

SOLUTION.  —  The  base   .     .     .     20  is  100%  of  20. 
The  percentage  .    _2  is     10%  of  20. 
Therefore  the  amount  .     22  is  110%  of  20.     Ans. 

4.  100%  +  10%  =  ?     1  +  10%  =  ? 

5.  If  20  is  diminished  by  10%  of  itself,  the  difference 
is  18.     What  per  cent  of  20  is  18  ? 

SOLUTION.  —  The  base  ...  20  is  100%  of  20. 
The  percentage  .  _2  is  10%  of  20. 
The  difference  .  "li  is  90%  of  20.  An*. 

6.  100%  -  10%  =  ?     1  -  10%  =  ? 

7.  What  number  increased  by  10%  of  itself  equals  220  ? 

SOLUTION. — Since  220  is  10%  more  than  the  required  number, 
220  is  110%  of  the  required  number. 

The  amount,  220,  is  now  treated  as  the  percentage,  and  110%  as 
the  rate;  and  the  question  becomes,  220  is  110%  of  what  number  ? 
(Question  6.) 

220  -f  1.10  =  200.     Ans. 

Amount  -f  (1  +  rate)  =  Base. 

-8.  What  number  diminished  by  10%  of  itself  equals 
180? 

SOLUTION. — Since  180  is  10%  less  than  the  required  number, 
180  is  90%  of  the  required  number. 

The  difference,  180,  is  now  treated  as  the  percentage,  and  90%  as 
the  rate ;  and  the  question  becomes,  180  is  90  %  of  what  number  ? 
(Question  6.) 

180  -f  .90  =  200.     Ans. 

Difference  -I-  (1  —  rate)  =  Base. 

9.    What  number  increased  by  25%  of  itself  equals  290  ? 
10.    What  number  diminished  by  25%  of  itself  equals 
243? 


PERCENTAGE.  123 

11.  After  selling  20%  of  his  sheep,  a  farmer  had  400 
sheep  left.     How  many  had  he  at  first  ? 

12.  The  population  of  a  certain  city  has  increased  12% 
in  two  years.     If  it  now  numbers  56000,  what  was  it  at 
the  beginning  of  the  two  years  ? 

13.  A   clerk's    salary  was  increased  6|%.     If  he    now 
receives  $850,  what  was  his  original  salary  ? 

14.  By  selling  goods  at  $630  I  lose  12i%.     What  did 
I  pay  for  them  ? 

15.  $580  is  10%  less  than  what  number? 

16.  I  sold  goods  at  $450,  which  was  120%  of  the  cost. 
What  was  the  cost  ? 

17.  After  withdrawing  45%  of  my  money  from  the  bank, 
I  still  have  $1300  on  deposit.     How  much  had  I  in  the 
bank  at  first  ? 

18.  A  farmer  increased  his  flock  of  sheep  by  12^%,  and 
then  had  900.     How  many  had  he  at  first  ? 

19.  A  man,  after  spending  a  month  in  the  Adirondacks, 
finds  that  his  weight  is  210  pounds,  which  is  an  increase  of 
5%.     What  was  his  weight  before  he  went  to  the  Adiron- 
dacks ? 

^  20.  A  regiment  lost  12i%  of  its  men  in  an  engagement, 
and  had  560  left.  How  many  men  were  there  before  the 
engagement  ? 

21.  A  owes  C  33^%  more  than  he  owes  B.     If  he  owes 
C  $800,  how  much  does  he  owe  B  ? 

22.  1227.83  is  1%  less  than  what  number? 

23.  |  is  20%  less  than  what  number? 

24.  A  city  lot  cost  $3600,  which  is  55%  less  than  the 
cost  of  the  house.     What  was  the  cost  of  the  house  ? 


124  SENIOR    ARITHMETIC. 

25.  A  farmer  raised  1500  bu.  of  corn,  which  was  33*% 
less  than  the  number  of  bushels   of  wheat  raised.     How 
many  bushels  of  wheat  had  he  ? 

26.  In  the  year  1896  a  merchant's  profits  were  $1836.24, 
which  was  25%  more  than  his  profits  of  1895.    What  were 
his  profits  in  1895  ? 

PROFIT    AND   LOSS. 

242.  Oral. 

State  the  question  only. 

1.  How  much  is  a  10%  profit  on  goods  that  cost  $200  ? 

2.  T  bought  goods  for  $400,  and  sold  them  at  a  loss  of 
5%.     How  much  did  I  lose  ? 

3.  If  I  buy  goods  at  $400,  and  sell  them  at  $600,  what 
per  cent  profit  do  I  make  ? 

4.  If  I  buy  at  $400,  and  sell  at  $350,  what  %   do  I 
lose? 

5.  By  selling  a  house  for  $1600  I  gain  33  J%.     What 
was  the  cost  ? 

6.  John  sold  his  skates  for  64  cents,  and  thereby  lost 
5  % .     What  did  he  pay  for  them  ? 

243.  All   computations  in  Profit  and  Loss  come  under 
the  rules  of  Percentage. 

The  cost  corresponds  to  the  base,  and  the  gain  or  loss 
is  a  percentage  of  the  cost. 

The  selling  price  is  the  amount  when  there  is  a  profit, 
and  the  difference  when  there  is  a  loss. 

244.  Written. 

7.  If  I  buy  eggs  for  10  cents  a  doz.,  and  sell  them  for 
12J  cents,  what  per  cent  do  I  gain  ? 

8.  A  grocer  bought  tea  at  18  cents  per  pound,  and  sold 
it  at  30  cents  per  pound.     What  was  the  rate  of  gain  ? 


PERCENTAGE.  125 

9.    Find  the  profit  on  a  bicycle  that  cost  $75,  and  was 
sold  at  an  advance  of  30%. 

10.  Find  the  selling  price  of  a  horse  bought  at  $88.65, 
and  sold  at  3J%   below  cost. 

11.  Find  the  rate  per  cent  of  loss  on  a  cow  bought  for 
$80,  and  sold  for  $60. 

12.  Find  the  rate  per  cent  profit  on  a  car-load  of  Cort- 
land  wagons  sold  for  $1090,  and  bought  for  $1000. 

13.  Find  the  cost  of  a  herd  of  cattle  sold  at  12±-%  above 
cost  at  a  profit  of  $240. 

14.  A  man  bought  books  for  $194,  and  sold  them  at  a 
gain  of  32  % .     What  was  the  gain  ? 

15.  I  sold  a  house  and  lot  that  cost  $11225  at  a  loss  of 
51%.     What  was  the  loss  ? 

16.  Mr.  A.,  by  selling  his  horse  at  a  profit  of  14%,  made 
$32.20.     What  did  the  horse  cost  ? 

17.  By  selling  sugar  at  one-half  cent  per  pound  profit,  a 
grocer  makes  ten  per  cent.     What  does  he  get  per  pound 
for  his  sugar  ? 

18.  An  agent  gained  $.09  by  selling  twine  25%  above 
cost.     What  did  it  cost  him  ? 

19.  Find  the  cost  of  cotton  sold  at  16-2%  above  cost  at 
a  profit  of  $211.25. 

20.  By  selling  flour  at  a  loss  of  14f  %,  a  grocer  loses 
$13.45.     What  was  the  cost? 

21.  A  farm  that  cost  $2675  was  sold  for  $3745.     What 
was  the  gain  per  cent  ? 

22.  Hats  that  cost  $43.50  a  doz.   are  sold   for  $4.50 
apiece.     What  is  the  rate  of  gain  ? 

23.  By   selling  boots   for  $206.40    a   merchant   gained 
20%.     What  did  they  cost  him? 


126  SENIOR    ARITHMETIC. 

24.  By  selling  corn  for  $92.61,  a  man  gained 
What  did  it  cost  him  ? 

25.  I  sell  a  horse  for  twenty  per  cent  less  than  my  ask- 
ing price,  and   yet  make  twenty-five  per  cent  profit.      I 
asked  $200.     What  did  the  horse  cost  me  ? 

26.  My  height  is  6  feet  1J  inches,  my  neighbor  is  5  feet 
10  inches.     What  per  cent  am  I  taller  than  he  is  ? 

27.  A  farmer  sold  160  acres  of  land  for  $2944,  which 
was  8%  less  than  it  cost.     What  did  it  cost  an  acre  ? 

28.  By  selling  a  horse  for  $160,   I  lose  20%.     What 
would  have  been  the  selling  price  had  I  gained  20%  ? 

29.  14f%  was  gained  by  selling  tea  at  $.45  a  pound. 
What  did  it  cost  a  pound  ? 

30.  Mr.  Brown  sold  a  lot  for  $4300,  and  by  so  doing 
made  11^%.     What  did  he  gain? 

31.  If  I  buy  oranges  at  the  rate  of  3  for  3  cents,  and 
sell  them  at  the  rate  of  2  for  5  cents,  what  per  cent  profit 
do  I  make  ? 

32.  A  jeweller  sold  two  watches  at  $24  each.     On  one 
he  gained  20%,  and  on  the  other  lost  20%.     What  did 
both  watches  cost  him  ? 


COMMISSION. 

245.    Oral. 

1.  A  certain  agent  receives  for  his  services  2%  of  the 
value  of  the  goods  which  he  sells.      How  much  will  he 
receive  for  selling  $1000  worth  of  goods  ? 

2.  A  purchasing  agent  receives  for  his  services  3%  of 
the  value  of  goods  purchased.     How  much  will  he  receive 
for  purchasing  $2000  worth  of  goods  ? 

3.  How  much  must  I  pay  my  agent  for  selling  $3000 
worth  of  potatoes  if  I  pay  him  5%  ? 


PERCENTAGE.  127 

4.  At  5%    how  much  will  a  collecting  agent  receive 
for  collecting  $800  ? 

5.  How  much  must  I  pay  my  broker  for  selling  $1000 
worth  of  stocks,  if  I  pay  him  £%  of  their  value  ? 

246.  An  Agent  is  a  person  who  transacts  business  for 
another. 

247.  Some  agents  are  known  as  Brokers,  or  Commission 
Merchants,  according  to  the  kind  of  business  transacted. 

248.  The  compensation  of   an  agent  is  called  Commis- 
sion, or  Brokerage. 

The  commission  of  a  purchasing  agent  is  usually  a  cer- 
tain per  cent  of  the  value  of  his  purchases. 

The  commission  of  a  sales  agent,  or  of  a  collector,  is 
usually  a  certain  per  cent  of  the  amount  collected. 

249.  The  merchandise  sent  to  a  commission  merchant  to 
be  sold  is  called  a  Consignment. 

250.  The   sender   is  the  Consignor,   and  the    person  to 
whom  the  goods  are  sent  is  the  Consignee. 

251.  The  commission  is  the  percentage,  and  the  amount 
collected  or  invested  is  the  base. 

252.  Written. 

6.  An  auctioneer  charges   5%    commission  for  selling 
$864  worth  of  goods.     What  is  the  amount  of  his  com- 
mission ? 

7.  Sold  850  barrels  of   flour  at  $5.25  a  barrel,  and 
charged  2|%   commission.     Find  my  commission. 

8.  What  is  an  agent's  commission  for  selling  6840  Ib. 
of  butter,  at  19  cents  a  pound,  commission  \\%  ? 

9.  A  dealer  sells  real  estate  for  a  commission  of  2%. 
How  much  must  he   sell  during  the   year  to    secure   an 
income  of  $75  per  month? 


128  SENIOR    ARITHMETIC. 

10.  A  broker  in  New  York  received  ^  of  one  per  cent 
commission  for  negotiating  a  sale   of   500  one  thousand 
dollar  bonds.     What  was  his  commission  ? 

11.  A  real  estate  man  made  $50   by  receiving  2^  per 
cent   instead    of   his   regular  commission    of   2    per  cent. 
What  did  his  sales  amount  to  ? 

12.  An   agent's   fee  for  collecting   bills  is   3%.     If  he 
receives  $86.25  as  his  commission,  how  much  money  has 
he  collected  ? 

13.  An    agent   collected    $1864    from    a   sale   of    some 
pictures,  and  received  $4.66  as  his  fee.     What  was  the 
rate  of  commission  ? 

14.  An  agent  having  sold  1250  velocipedes  at  $8  apiece, 
invested  his  commission  of  If  %  in  a  new  stock  company. 
How  many  shares  at  $25  each  did  he  take  ? 

15.  What  per  cent  does  an  agent  charge  who  receives 
$223  for  buying  $5575  worth  of  produce  ? 

16.  A  man  sejids  his   agent   $6000   to  invest  in   flour, 
after  deducting  his  commission  of  2%.     How  much  money 
is   spent   for  the    flour,   and    how   much   for   the    agent's 
commission  ? 

17.  A  man   is   paid   5%    for  collecting  $235.75.     How 
much  must  he  pay  over  to  his  employer  ? 

18.  A  merchant  sent  his  agent  $3150  with  which  to  buy 
flour  after  deducting  his  commission  of  5%.     At  $4  per 
barrel  how  many  barrels  did  the  agent  buy  ? 

19.  An  agent  sold  iron  for  $9872.     He  received  $163.70, 
which  included  a  freight  charge  of  $52.64.     What  rate  of 
commission  did  he  receive  ? 

20.  Received    as   net    proceeds    from   a  sale   of   cotton 
$1025.70,  after  paying  my  agent  1\%  for  selling.     What 
did  the  sale  amount  to  ? 


PERCENTAGE.  129 

21.  An  auctioneer  sells   15  tables  at  $1.45  apiece,  22 
chairs  at  $1.121  apiece,  and  some  pictures  for  $8.70,  on 
a  commission  of  o\°/0.     What  was  his  commission,  and  the 
net  proceeds  of  the  sale  ? 

22.  My  agent  in  Boston  sold  a  number  of  bicycles  at 
$85  each.     After  deducting  his  commission  of  3£%,  he  re- 
turned to  me  $5759.60.     How  many  bicycles  did  he  sell  ? 

23.  An  agent  who  sold  150  lots  at  $233^  each,  charged 
$262.50  for  his  services.     What  rate  of  commission  did  he 
get? 

24.  A  collector  pays  over  to  his   principal  $23358.39^ 
after  deducting  a  commission  of  4£%.     How  much  was  the 
entire  collection  ? 

25.  If  I  send  my  agent  $367.20,  with  instructions  to  buy 
tea  at  30  ct.  a  pound,  and  he  charges  2%  for  buying,  how 
many  pounds  of  tea  should  I  receive  ? 

26.  A  real  estate  agent  charges  me  two  per  cent  for  sell- 
ing my  property  in  Boston.     He  remits  me  $5880.     What 
was  his  commission  ? 

27.  A  commission  merchant  _in  New  York  charged  $36 
for  insuring  my  goods,  $14  for  cartage,  and  $50  commis- 
sion at  2i  per  cent  for  selling  them.     How  much  money 
should  he  remit  to  me  ? 

--  28.  Sent  my  agent  $2050  to  invest  in  coal  at  $4  per  ton, 
after  deducting  his  commission  of  2i  per  cent.  How  many 
tons  of  coal  could  he  buy  ? 

29.  A  cotton  broker  received  $2531.71  with  which  to  buy 
cotton  at  $.12  a  Ib.     He  charged  2-J%  commission.     How 
many  pounds  of  cotton  did  he  buy,  and  what  was  his  com- 
mission ? 

30.  A  real  estate  agent  receives  $162,193.50  from  a  com- 
pany to  invest  in  land.     If  he  charges  5%  commission,  how 


130  SENIOR    ARITHMETIC. 

many  acres  of  land  can  he  buy  at  $9  an  acre  ?     What  is 
his  commission  ? 

31.  An    agent   sold  12000   Ib.  of   cotton  at  10  cents  a 
pound.     He  invested  the  proceeds  in  lumber  at  $25  per  M. 
If  his  commission  for  selling  was  4%,  and  for  buying  2%, 
how  many  feet  of  lumber  did  he  purchase  ? 

32.  A  grain-dealer  received  $4820.40  with  which  to  buy 
wheat  at  60^  a  bushel  after  deducting  his  commission  of 
3%.     How  much  wheat  did  he  purchase  ? 

33.  How  much  stock  can  be  bought  for  $10827,  allowing 
li%  brokerage  ? 

INSURANCE 

253.  1.    I  keep  my  house  insured  for  $4000.     I  pay  the 
insurance   company  1%   annually.      How  much  do  I  pay 
annually  ? 

2.  If  my  house  (Ex.  1)  burns  down  at  the  end  of  three 
years,  how  much  shall  I  receive  from  the  company  more 
than  I  have  paid  them  ? 

3.  If  I  pay  $75  per  annum  for  insuring  my  house  at 
1%,  for  how  much  is  it  insured  ? 

4.  At  2%   what  will  be  the  cost  of  insuring  $10000 
of  merchandise  at  j  value  ? 

254.  Insurance  is  security  against  loss. 

255.  The  payment  made  for  insurance  is  called  the  Pre- 
mium.    Some  of  the  different  kinds  of  insurance  are, 

256.  Fire,  Lightning,  Tornado,  Accident,  Life,  and  Marine. 

257.  The  contract  between  the  insurer  and  the  insured 
is  called  the  Policy. 

258.  The  amount  insured  is  the  base,  and  the  premium 
is  the  percentage. 


PERCENTAGE.  131 

-  5.    What  will  it  cost  to  insure  a  house  worth  $ 2500  at 
|  of  its  value,  for  three  years,  at  f  %  ? 

6.  Insured  a  country  store   for  $5,000  and   goods  for 
$10,000,   at   30X   on    $100.      $1   is  paid  for  the  policy. 
What  does  the  insurance  cost  ? 

7.  What  will   it  cost  to  insure  a  mill  for  $5000,  the 
rate  being  one  and  one-half  per  cent  for  3  years  ? 

8.  How  much  will  I  save  by  insuring  my  property  for 
$5000  at  |  of  one  per  cent  for  3  years,  rather  than  taking 
an  annual  policy  for  |  of  one  per  cent  ? 

9.  It  costs  me  to  insure  my  house  $22.50  when  the  rate 
is  |  of  one  per  cent.     What  is  the  amount  of  my  policy  ? 

10.  A  stock  of  goods  is  insured  for  one-half  the  value, 
the  premium  being  $30,  and  the  rate  T%  of  one  per  cent. 
What  is  the  value  of  the  goods  ? 

11.  The  semi-annual  premium  per  one  thousand  dollars 
on  my  $6000  life-insurance  policy  is  $26.     What  does  it 
cost  me  a  year  ? 

12.  A  person  who  pays  $12  semi-annually  for  accident 
insurance  is  disabled  by  an  accident  for  13  weeks,  during 
which  time  he  receives  $10  a  Aveek.     If  he  has  paid  three 
premiums,  how  much  more  does  he  receive  than  he  has 
paid  out  ? 

13.  Paid  for  insuring  a  house  for  §  of  its  value,  $151. 
The  rate  being  75^  on  $100,  and  the  policy  costing  $1, 
what  was  the  house  worth  ? 

14.  To  insure  a  house  at  ^  of  1%  cost  me  $20.     What 
was  the  house  worth  ? 

15.  Paid  $18  for  insuring  goods  worth   $9000.     What 
was  the  rate  ? 

16.  A  merchant  pays  $75  a  year  insurance  on  his  stock 
of  goods  at  1  \%.    What  is  the  value  of  his  stock  of  goods  ? 


132  SENIOR   ARITHMETIC. 

17.  A  block  worth  $30000  is  insured  for  §  of  its  value 
at  2%.     How  much  does  the  owner  lose  in  case  of  its 
total  destruction  by  fire  ? 

18.  For  how  much  must  a  cargo  of  wheat  worth  $24000 
be  insured,  at  2i%,  so  that  the  owner,  in  case  of  loss,  may 
recover  both  the  value  of  the  cargo  and  the  premium. 

NOTE.  —  The  value  of  the  wheat  =  97£%  of  the  amount  insured. 

TRADE   DISCOUNT. 

259.  The  deduction  of  a  percentage  from  the  price  of 
merchandise  is  called  Commercial  Discount. 

It  is  used  largely  by  manufacturers  and  wholesale  mer- 
chants. The  greatest  discounts  are  for  large  purchases 
and  cash  payment. 

260.  The  List  Price  is  the  price  given  in  the  price-list. 

261.  The  Net  Price  is  the  list  price  -less  the  discount. 

1.  If  I  can  purchase  books  at  25%  off  for  cash,  what 
must  I  pay  for  books  listed  at  $80  ? 

SOLUTION.  —  100%  —  25%  =  75%.     75%  of  $80  =  $60.     Ans. 

NOTP:.  —  When  two  or  more  discounts  are  allowed,  the  first  is 
deducted,  the  second  computed  on  the  remainder,  and  deducted  from 
it,  etc. 

2.  At  what  per  cent  above  cost  must  a  merchant  mark 
his  goods  so  that  he  may  allow  a  discount  of  25%  from 
the  marked  price,  and  still  make  a  profit  of  10%  ? 

SOLUTION.  —  Selling  price  =  110%  of  cost.  This  selling  price  is 
75%  of  the  marked  price.  The  question  is,  110%  is  75%  of  what. 
1.10  -7-  75  =  1.46|%,  therefore  the  marked  price  is  46|%  above  cost. 

3.  Find  the  sum  to  be  paid  on  a  bill  of  $264  with  10% 
off  for  cash. 

4.  What  is  the  net  price  of  a  bill  of  goods,  the  list 
prict1  of  which  is  $56,  subject  to  discount  of  25%  ? 


PERCENTAGE.  133 

5.  What  must  be  paid  on  $935,  if  15%   and  10%   off 
are  allowed  ? 

SOLUTION.  —  Deducting  15$  is  the  same  as  allowing  85$  of  the 
bill.     85$  of  935  =  794.75.     90$  of  794.75  =  715.28.     Ana. 

6.  Which   is   the    better  for  the  buyer,  40%,  or  25% 
and  15%  off? 

7.  Find  a  single  discount  on  a  bill  of  $300  equal  to 
20%  and  5%  off. 

8.  A  discount  of  $4  was  allowed  on  a  bill,  which  was 
then  paid  with  a  check  for  $36.     What  rate  per  cent  was 
taken  off  ? 

9.  Consulting  niy   price-list,  I   find   I  can    buy  goods 
which  are  marked  $450  at  a  discount  of  20%  and  5%  off 
for  cash.     How  much  will  the  goods  cost  me  ?   and  how 
much  discount  do  I  receive  ? 

10.  Bought  furniture  amounting  to  $520  on  credit  for 
6  months,  or  5%   discount  for  cash.     What  ready  money 
will  pay  the  bill  ? 

11.  What  is  the  cash  value  of  a  bill  of  books  amounting 
to  $40,  on  the  face  of  which  a  discount  of  20%  and  5%  is 
made  ? 

12.  The  net  amount  of  a  bill  of  goods  is  $359.10.   What  is 
the  gross  amount,  the  rate  of  discount  being  10%  and  5%  ? 

13.  A  set   of  Encyclopaedias,  whose   catalogue  price  is 
$100,  can  be  bought  at  a  discount  of  2  tens  and  5%  off  for 
cash.      How  much  less  than  the  catalogue  price  will  they 
cost? 

NOTE.  — The  expression  2  tens  and  5$  means  10$,  10$,  and  5$. 

14.  B  offers  me  some  hammocks  for  $450  with  a  dis- 
count of  20%,  and  4%  off  for  cash;  and  A  offers  me  the 
same  goods  at  a  discount  of  2  tens  and  4%  off.     Which  is 
the  better  offer  ?  and  how  much  ? 


134  SENIOR   ARITHMETIC. 

15.  A  dealer  sold  goods  at  10%  below  his  asking  price, 
but  still  made  a  profit  of  20%.     What  per  cent  above  cost 
had  he  marked  the  goods  ? 

16.  A  merchant  marked  carpeting  that  cost  him  60  cents 
a  yard  so  that  he  could  allow  a  discount  of  10%  and  still 
make  a  profit  of  20  % .     At  what  price  did  he  mark  it  ? 

17.  A  book-dealer  sold  a  stock  of  books  for  $1140,  at  a 
discount  of  10%  from  the  marked  price,  and  finds  that  he 
has  made  a  profit  of  14%.    What  did  he  pay  for  the  books  ? 
and  what  was  their  marked  value  ? 

18.  Find  the  net  amount  of  a  bill  for  $386  subject  to 
the  following  discounts,  20%,  10%,  and  5%. 

TAXES 

262.  A  tax  is  a  sum  of  money  levied  upon  property  and 
persons  for  public  use. 

NOTE.  —  A  tax  upon  persons  is  called  Capitation  or  Poll  Tax. 
It  is  levied  in  some  localities  upon  men  of  full  age,  without  regard 
to  their  property.  It  is  usually  but  a  small  amount  upon  each 
person.  The  practice  is  going  out  of  use, 

263.  Property  is  of  two  kinds,  Real  and  Personal. 

264.  Real   Property  is   immovable    property,   as    lands 
and  buildings. 

265.  Personal  Property  is  property  that  is  movable,  as 
money,  securities,  household  goods,  horses,  cattle,  etc. 

266.  A  tax  assessed  upon  property  is  a  Property  Tax. 

267.  Assessors  are  officers  chosen  to  make  a  list  of  tax- 
able property,  estimate  its  value,  and  apportion  the  tax. 

268.  A  tax  is  a  percentage  upon  the  assessed  valuation 
of  property.     The  tax  on  $1  is  the  rate. 

1.    The    valuation   of    property   in   a   certain   town   is 
$1,500,000,  and  the  rate  is  11%.     What  is  the  tax? 


PERCENTAGE.  135 

2.  The  tax  to  be  raised  in  a  certain  village  is  $37500. 
The  taxable  property  is  $2,500,000.     What  is  the  rate? 
What  will  be  A's  tax  on  $15000  real  estate,  and  $3000 
personal  ? 

3.  The  property  of  a  town  is  assessed  at  $1,250,000. 
The  tax  to   be   raised  is   $15975.     There  are  650   polls, 
assessed  at  $1.50   each.     What  is   B's   entire  tax,  if  his 
property  is   assessed    at   $2500,   and   he    pays    the   poll- 
tax  ? 

Rule.  —  Deduct  the  amount  of  poll-tax,  If  any,  from  the 
whole  tax.  Divide  the  remainder  by  the  assessed  val- 
uation. The  quotient  will  be  the  rate. 
To  find  each  person's  tax,  multiply  the  assessed  valua- 
tion by  the  rate,  and  to  the  product  add  the  poll-tax, 
If  any. 

4.  The  officers  of  a  certain  town  find  that  all  the  town 
expenses  for  the  year  1896  will  amount  to  $46000.     The 
tax-roll  shows  real  estate  valued  at  $2,000,000,  and  per- 
sonal property  at  $300000.     What  is   the   rate  of  taxa- 
tion ? 

— -5.  A  certain  town  votes  to  raise  a  tax  of  $14250,  be- 
sides the  collector's  commission  of  5%.  What  is  the  rate 
of  taxation  if  the  property  valuation  is  $1,000,000  ? 

What  is  the  collector's  commission,  and  what  is  A's  tax. 
on  property  valued  at  $4,500  ? 

6.  If  the  assessed  valuation  of  a  village  is  $2,384,564, 
and  there  are  750  polls  at  $1.50  each,  what  must  be  the 
rate  of  taxation  to  meet  an  expense  of  $29807.05  ?     What 
is  B's  entire  tax,  if  his  property  is  valued  at  $3875,  and  he 
pays  for  1  poll  ? 

7.  What  is  the  valuation  of  my  property,  if  my  tax,  15 
mills  on  a  dollar,  amounts  to  $30. 


136  SENIOR    ARITHMETIC. 

8.  What  is  my  entire  tax,  if  I  pay  a  poll-tax  of  $1.68, 
and  my  property  is  valued  at  $24750,  when  the  rate  of 
taxation  is  $16.28  on  $1000? 

9.  The  annual  tax-rate  for  the  State  of  New  York  for 
the  year  1896  was  2.69  mills  on  the  dollar.     The  amounts 
to  be  raised  by  tax  are  as  follows:  $961116  for  general 
expenses,  $4,062,903  for  free  schools,  $2,360,103  for  the 
canals,  and  $4,368,712  for  the  State  care  of  the  insane. 
What  was  the  assessed  valuation  of  the  property  of  the 
entire  State  ? 

The  tax-rate  for  1895  was  3.24  mills.     What  was  the 
entire  tax  of  1895? 

DUTIES. 

269.  Duties  are  taxes  on  imported  goods,  levied  by  the 
government,  and  collected  at  custom-houses.     A  port  con- 
taining a  custom-house  is  called  a  Port  of  Entry. 

270.  An  Ad  Valorem  Duty  is  a  certain  rate  on  the  value 
of  goods  at  the  place  from  which  they  were  shipped. 

271.  A  Specific  Duty  is  a  fixed  sum  charged  upon  an  im- 
ported article,  without  regard  to  its  value.      Allowances 
are  made  as  follows,  in  collecting  specific  duties :  for  Tare, 
which  is  weight  of  box,  cask,  etc.  ;  for  Leakage,  which  is 
loss  of  liquids  in  barrels  or  casks ;  and  for  Breakage,  which 
is  loss  of  liquids  in  bottles. 

272.  The  Gross  Weight  is  the  weight  of  articles  before 
any  allowances  are  made. 

273.  The  Net  Weight  is  the  weight  after  the  allowances 
are  made. 

1.  At  30  per  cent  ad  valorem,  what  is  the  duty  on  goods 
valued  at  $725  ? 

2.  What  is  the  duty  on  10  gross  of  silver  spoons,  val- 
ued at  $4.50  a  dozen,  at  30%  ad  valorem  ? 


PERCENTAGE.  137 

3.  A.  Mark's  Sons  imported  from  Lyons  1560  yd.  of 
silk  invoiced  at  87 \fl  per  yard.     What  was  the  duty  at 
25/  a  yard,  and  30^  ad  valorem? 

4.  If  the  average  rate  of  duty  under  the  McKinley  law 
was  49.58  per  cent,  and  under  the  Wilson  law  it  is  37  per 
cent,  what  is  the  difference  in  revenue  on  $1,000,000  worth 
of  dutiable  imports  ? 

5.  A  merchant  bought  goods   in   London  invoiced   at 
£450.     At  the  custom-house  in  New  York  he  paid  an  ad 
valorem  duty  of  18%,  and  a  specific  duty  of  $325.    What 
was  the  entire  cost  of  the  goods  in  United  States  money  ? 

6.  Imported  from  England  5  cases  of  cloths  and  cash- 
meres, net  weight  95  Ib.  ;  value  as  per  invoice  £375  10s. 
What  is  the  duty,  the  rate  being  50^  per  pound,  and  35% 
ad  valorem  ? 

QUESTIONS. 

274.  1.    What  is  the  meaning  of  the  term  per  cent  ? 
How  is  per  cent  written  ? 

2.  Define  base,  percentage,  rate  per  cent,  amount,  dif- 
ference. 

3.  Tell  how  to  find  percentage  when  base  and  rate  are 
given.     To  find  base  when  percentage  and  rate  are  given. 
To  find  rate  when  percentage  and  base  are  given. 

4.  Tell   how  to  find  base  when  amount  and  'rate  are 
given.     When  difference  and  rate  are  given. 

5.  Define  Commission,  Brokerage,  Insurance,  Premium, 
Policy,  Taxes,  Real  Estate,  Personal  Property. 

6.  What  is  trade  discount  ?     List  price  ?    Net  price  ? 
Give  rule  for  finding  net  price. 

MISCELLANEOUS   REVIEW   OF   PERCENTAGE. 

275.  l.    Find  8%  of  750.     61%  of  $12.75.     |%of912. 
of  2140. 


138  SENIOK   AKITHMETIC. 

2.  A  man  gave  his  son  42%  of  his  money,  his  daughter 
25%  of  it,  and  his  wife  16§%   of  the  remainder.     If  the 
son  received  $9350  more  than  the  daughter,  what  did  each 
receive  ? 

3.  A  dealer  sold  a  horse  and  carriage  for  $637,  which 
was  40%  more  than  cost.     If  the  horse  cost  f  as  much  as 
the  carriage,  what  did  each  cost  ? 

4.  What  per  cent  is  gained  when  one-half  an   article 
is  sold  for  what  the  whole  cost  ?     When  f  of  an  article  is 
sold  for  what  one  half-cost  ? 

5.  A  merchant  pays  $35  for  a  suit  of  clothes.     What 
must  he  ask  for  it,   so  that  he  may  drop  16%   from  his 
asking  price,  and  still  make  20%  on  the  cost  ? 

6.  14.35  is  T7ff  %  of  what  number  ? 

7.  A  man  spent  20%  of  his  salary  for  board  and  15% 
of  what  was  left  for  clothes.     If  he  spent  $132  more  for 
board  than  for  clothes,  how  much  did  he  spend  for  each  ? 

8.  What  number  diminished  by  16§%  is  605  ? 

9.  What  number  increased  35%  is  382.5  ? 

10.  Sold  a  load  of  wheat  weighing  3240  Ib.  at  68^  a 
bushel  of  60  Ib.,  thereby  making  a  profit  of  6|  per  cent. 
Required  the  cost  of  the  wheat  ? 

11.  On  a  certain  day  the  sun  rose  at  5  o'clock  and  43 
minutes,  and  set  at  6  o'clock  and  25  minutes.     What  per 
cent  of  the  day  was  in  sunlight  ? 

12.  The    salary  of   a  certain    teacher   of   arithmetic   is 
$1600.     His  real  estate  tax  is  $90  ;  his  water  tax  is  $25  ; 
gas  bill,  $15  ;  coal  bill,  $45  ;  other  expenses,  $325.     What 
per  cent  of  his  salary  does  he  save  in  the  bank  ? 

13.  The  Oswego  Starch  Factory  employs  700  operatives. 
The  population  of  Oswego  numbers  22000.    What  per  cent 
of  the  population  is  employed  in  the  starch  factory  ? 


PERCENTAGE.  139 

14.  |  is  25  %  more  than  what  fraction  ? 

15.  A  dealer  lost  10%  of  his  capital,  then  gained  20% 
of  the  remainder,  when  he  had  $2160.     How  much  had  he 
at  first  ? 

16.  Goods  bought  for  $400   are   marked  to  sell  at  anj 
advance  of  40%,  but  are  finally  sold  at  a  reduction  of  25% 
from  the  marked  price.     What  is  the  per  cent  of  gain  ? 
What  is  the  gain  ? 

17.  An  article  is  sold  for  $2.80,  this  being  an  advance  of 
25%.     Find  the  cost. 

18.  A  merchant  buys  sugar  at  an  average  price  of  4 
cents  a  pound,  and  sells  at  a  profit  of  8%.     How  many 
pounds  must  he  sell  to  clear  $500  ? 

19.  If  by  selling  an  article  for  59  cents  a  dealer  gains 
10%  more  than  by  selling  for  55  cents,  what  is  the  original 
cost  ? 

20.  15%  of  an  estate  is  invested  in  city  bonds,  40% 
in  real  estate,  25%  in  railroad  stock,  and  the  remainder, 
$5000,  is  deposited  in  a  bank.     WThat  is  the  estate  worth  ? 

21.  Define  percentage,  base,  profit  and  loss,  commission. 

22.  Give  the  five  formulas  of  percentage. 

23.  Express  as  a  decimal  |  per  cent. 

24.  A  man  having  a  yearly  income    of  $1500   spends 
80%  of  it  the  first  year,  75%  of  it  the  second  year,  62  £% 
of  it  the  third.     How  much  does  he  save  in  3  years  ? 

25.  25%  of  200  bushels  is  2£%  of  how  many  bushels  ? 

26.  A  man  sold  80  acres  of  land  for  $1472,  which  was 
8%  less  than  it  cost.     What  did  it  cost  an  acre  ? 

27.  What  terms  in  Profit  and  Loss  correspond  to  base 
and  amount  ? 

28.  Find  the  cost  of  fruit  sold  for  $207.70,  at  a  gain  of 


140  SENIOR    ARITHMETIC. 

29.  At  what  price  must  hats  that  cost  $1.12  each  be 
marked  in  order  to  abate  5  % ,  and  yet  make  25  %  profit  ? 

30.  What  is  the  base  in  commission  ? 

31.  A  commission  merchant  sells  225  bu.  of  corn  at  $.65 
a  bushel,  and  360  bbl.  of  apples  at  $2.40  per  barrel ;  com- 
mission 5%.     Find  the  commission  and  the  net  proceeds. 

32.  The  net  proceeds  are  $3800,  the  rate  10%.     Find 
the  amount  of  sales  and  the  commission. 

33.  Find  the  rate,  the  commission  being  $125,  and  the 
sum  invested  $2500. 

34.  A  merchant  owning  f  of  a  cargo  valued  at  $44000  in- 
sures |  of  his  share  at  2|%.    What  premium  does  he  pay  ? 

35.  A  man  having  $400  paid  62i%  of  it  for  a  carriage. 
How  many  dollars  had  he  left  ? 

36.  An    agent    charged    $432.46    for    selling    goods   at 
$49424.     WThat  was  his  rate  of  commission  ? 

37.  A  man  sold  four  horses  for  $100  each ;  on  two  he 
gained  25%,  and  on  the  other  two  he  lost  25%.     Did  he 
gain  or  lose  on  the  transaction  ?  and  how  much  ? 

38.  If  f  the  number  of  girls  in  a  certain  school  exceed 
the  boys  10%,  and  the  girls  number  275,  what  is  the  num- 
ber of  boys  ? 

39.  A  farmer's   sheep  increased  10%   each  year  for  2 
years,  when  he  had  242.     How  many  had  he  at  first  ? 

40.  My  New  York  agent  buys  for  me  40  pieces  of  silk, 
32  yards  in  a  piece,  at  $5  a  yard.     He  charges  li%  com- 
mission.    How  much  money  will  it  require  to  purchase  the 
silk  and  pay  his  commission  ? 

41.  A  commission  merchant  in  Boston  has   sold   goods 
for  me  to  the  amount  of  $6932.     He  has  charged  1|% 
commission,  $18.50  cartage,  and  $12.15  for  storage.     How 
much  is  due  me  ? 


PERCENTAGE.  M-O 

42.  A  boy  bought  oranges  at  the  rate  of  3  for  5  cents, 
and  sold  them  at  the  rate  of  2  for  5/.  What  was  his  rate 
of  gain  ? 

-  43.    Ten  per  cent  of  a  number  is  32  less  than  eighteen 
per  cent  of  the  same  number.     What  is  the  number  ? 

44.  I  paid  $28.87^-  for  insuring  my  house  for  $3850  for 
three  years.     What  was  the  rate  of  the  yearly  premium  ? 

45.  A  stock  of  goods  valued  at  $6300  is  insured  for  § 
its  value  at  f  %.     WThat  will  be  the  owner's  loss  if  ,the 
goods  are  totally  destroyed  by  fire  ? 

46.  A  man's  income  is  $1720,  which  is  16f  %  of  the  sum 
he  has  invested.     What  sum  has  he  invested  ? 

47.  From  a  cargo  containing  wheat,  1620  bu.,  or  7%,  was 
washed  overboard.     What  number  of  bushels  remained  ? 

48.  A  stock-dealer  sold  38  head  of  cattle,  which  was  4% 
of  his  entire  herd.     How  many  had  he  left  ? 

49.  In  an  orchard   containing  820  trees,  20%  of  them 
were  pear-trees,  and  the  remainder  were  plum-trees.     How 
many  plum-trees  were  there  in  the  orchard  ? 

50.  From  a  cask  of  wine  containing  65  gal.  all  but  15% 
was  sold.     How  many  gallons  were  sold  ? 

51.  In  a  school  containing  875  pupils  32%  of  them  are 
boys  and  the  remainder  girls.     How  many  girls  are  there  ? 

52.  There  is  a  loss  of  $500  on  a  house  and  lot  sold  for 
$5000.     What  is  the  per  cent  of  loss  ? 

53.  An  agent  reports  that  he   invested  the  money  re- 
mitted him  in  wheat,  which  he  sold  at  an  advance  of  15%  ; 
then  investing  the  proceeds  in  a  second  quantity,  he  was 
forced  to  sell  at  a  loss  of  12|%.     He  now  deducts  $100 
for  expenses  and  commission,  and  remits  $5333.75  to  his 
employer  as  the  balance  due  him.     Find  the  loss  to  the 
employer. 


142  SENIOR    ARITHMETIC. 

54.  $90  are  paid  as  premium  for  insuring  a  block  for 
three-fourths  of  its  value.  If  the  rate  of  insurance  is  |%, 
what  is  the  value  of  the  property  ? 

65.  New  York  State  has  a  population  of  5,998,000,  and 
New  York  City  has  1,515,000.  What  per  cent  of  the 
population  of  the  State  live  in  New  York  City  ? 

56.  During  the  war  of   1861-1865,   the   State  of  New 
York  paid  $40,000,000  in  bounties  to  her  volunteers.    Her 
population  at  that  time  was  (in  round  numbers)  4,000,000. 
What  was  the  average  cost  to  each  inhabitant  ? 

57.  If  I  sell  6  horses  for  what  8  horses  cost,  what  is  my 
rate  of  gain  ? 

58.  Sold  wheat  for  $73.541,  by  which  a  gain  of  15%  was 
made.     What   did   the  wheat   cost  ?    and  what   sum  was 
gained  ? 

59.  In  a  school  containing  1160  pupils,  638  are  girls  and 
the  remainder  are,  boys.     What  per  cent  are  boys  ? 

60.  A  hall  is  42  ft.  wide,  and  294  ft.  long.     What  per 
cent  of  the  length  is  the  width  ? 

....  61.  In  a  certain  battle  22f  %  more  than  ^  of  the  soldiers 
were  killed.  If  the  loss  was  110  men,  what  was  the 
original  number  ? 

62.  In  an  orange-grove  8j  %   of  the  trees  were  ruined 
by  frost.     If  1100   remained  uninjured,  how  many  were 
destroyed  ? 

63.  A  merchant  sold  a  lot  of  goods  for  $550,  thereby 
gaining  10%.     Find  the  cost  of  the  goods. 

64.  A  man  sold  a  watch  for  $32,  thereby  losing  20%  on 
the  cost.     Find  the  cost. 

65.  If  a  man  owns  66 §  per  cent  of  a  factory,  and  sells 
33^  per  cent  of  his  share  for  $1800,  what  is  the  value  of 
the  factory  ? 


SIMPLE    INTEREST.  143 

66.  A  sold  30%  of  his  steamship  to  B~;  B  sold  60%  of 
his  purchase  to   C;   C  sold  75%   of   his  share    to  D  for 
$27000.     What  was  the  value  of  the  vessel  ?     What  was 
each  one's  share  in  dollars  after  the  sales  had  been  made  ? 

67.  After  a  discount  of  30%  had  been  made  upon  the 
catalogue  price  of  a  book,  it  was  sold  for  $1.75.     What 
was  the  catalogue  price  ? 

68.  Bought  a  horse  for  $120,  and  sold  him  for  $135.' 
What  part  of  the  cost  was  the  gain  ?     What  per  cent  ? 

69.  Bought  tea  at  60  ct.  a  pound.     What  must  I  ask 
per  Ib.  so  as  to  abate  10%  and  still  make  a  profit  of  25%  ? 

70.  A  merchant's  profits  for  1895  were    $3402.84.     If 
they  were  6|%  less  than  in  1894,  what  were  they  in  1894  ? 

71.  In  one  week   John    solved   75    problems   correctly. 
If  he  failed  in  16§%  of  the  number  attempted,  how  many 
were  there  in  all  ? 

SIMPLE   INTEREST. 

276.    1.    I  borrow  $500  for  1  year,  and  at  the  end  of  the 
year  I  repay  the  money  and  6%  -for  the  use  of  it.     How 
much  do  I  pay  for  the  use  of  $500  ? 
— -2.    How  much  must  be  paid  for  the  use  of  $50  for  1 
year  at  5%  ?  at  7%  ? 

3.  How  much  at  5  %  per  annum  must  I  pay  for  the  use 
of  $1000  for  1  year  ?     For  3  years  ? 

4.  I  loan  James  Barnes  $500  at  6%.     At  the  end  of  2 
years  he  pays  me  in  full.     How  much  does  he  pay  me  ? 

Money  that  is  paid  for  the  use  of  money  is  called  Inter- 
est. The  money  for  the  use  of  which  interest  is  paid  is 
called  the  Principal,  and  the  sum  of  the  Principal  and 
interest  is  called  the  Amount. 

Interest  at  6%  means  6%  of  the  principal  for  1  year. 


144  SENIOR    ARITHMETIC. 

12  months  of  30  days  each  are  usually  regarded  as  a 
year  in  computing  interest. 

Oral. 

5.   What  is  the  interest  of  $100  for  3  years  at  6%  ? 

SOLUTION.  —  $100  Principal. 

.06  Rate. 

$6.00  Interest  for  1  year. 
3 


$18.00  Interest  for  3  years. 

6.  What  is  the  interest  of  $80  at  5%  for  2\  years  ? 

7.  What  is  the  interest  of  $1000  at  5%  for  2  yr.  6  mo.  ? 

8.  What  is  the  interest  of  $100  at  6%    for  1  year? 
For  1^  ?     For  2  yr.  6  mo?     For  3  yr.  3  mo.  ?     For  1  yr. 
6  mo.  ? 

When  the  time  does  not  include  days,  find  interest  as 
follows : 

Principal  X  Kate  X  Time  =  Interest. 

9.  What  is  the  interest  of  $297.62  for  5  yr.  3  mo.  at 


SOLUTION.—   $297.62 
.06 


NOTE.  —  Final    results    should 


$17.8572^  not  include  millg<     Mills  are  dig. 

— 44^43  regarded  if  less  than  5,  and  called 

892860  another  cent  if  5  or  more. 


$93.75      Ans. 
Find  the  interest  of : 

10.  $384.62  at  6  %  for  2  yr.     12.    $250.50  at  8  %  for  5  yr. 

11.  $463.75  at  7%  for  3  yr.     13.    $685.20  at  4%  for  6  yr. 

14.  $596.15  at  5%  for  2  yr.  3  mo. 

15.  $386.42  at  5£%  for  6  yr.  5  mo. 

16.  $950.16  at  10%  for  4i  yr. 

17.  $283.25  at  6%  for  2  yr.  8  mo. 


SIMPLE   INTEREST.  145 

Find  the  amount  of  : 

18.  $284.10  for  3  yr.  2.  mo.  at  7%. 

19.  $364.24  for  1  yr.  1  mo.  at  6%. 

20.  $282.50  for  5  yr.  9  mo.  at  5J%. 

21.  $298  for  4  yr.  3  mo.  at  6%. 

22.  $389  for  7  yr.  10  mo.  at  5%. 

23.  $894  for  6j  yr.  at  5J%. 

24.  A  man  buys  a  house  and  lot  for  $2800.     He  pays  f 
of  the  amount  in  cash,  and  the  remainder  after  1  yr.  4  mo. 
with  5%  interest.    Find  the  amount  of  the  second  payment. 

25.  Kequired  the  simple  interest  and  amount  of  $787.875 
for  7  yr.  7  mo.  at  7%. 

26.  Find  the  interest  on  a  note  for  $12500  for  three 
months  at  8%. 

27.  A  man  paid  his  city  tax  five  months  after  it  became 
due.     His  tax  was  $560.     In  accordance  with  city  ordi- 
nance, 1%  is  added  for  each  1  month  the  taxes  are  over- 
due.    He  pays  to  the  city  collector  of  taxes,  who  adds  5% 
collection  fee.     How  much  did  he  have  to  pay  ? 

THE   SIX  PER  CENT  METHOD. 

277.    By  the  6%  method  it  is  convenient  to  find  first  the 
interest  of  $1,  then  multiply  it  by  the  principal. 

1.  If  $.09  is  the  interest  of  $1  for  a  certain  time,  what 
is  the  interest  of  $2  for  the  same  time  ?  of  $10  ?  of  $25  ? 

2.  The  interest  of  $1  at  6%  for  a  certain  time  is  $.034. 
What  is  the  interest  of  $36.25  for  the  same  time  ? 


EXPLANATION.  —  The  interest  of  $36.25  is  36$fe  times  the  inter- 
est of  $1. 

At  Qf0  the  interest  of  $1  for  1  year     =      .     .     .     .     $.06 
for  1  month  =  fa  of  $.06    =  $.00^ 
for  1  day       =  &  of  $.00^  =  -$.0004 


146  SENIOIt   ARITHMETIC. 

3.   What  is  the  interest  of  $50.24  at  6%  for  2  yr.  8 
mo.  18  da.  ? 
SOLUTION.  — 

The  interest  of  $1  for  2  yr.  =  2  x  8.06  =  $.12 
for  8  mo.  =  8  x  $.00^  =  .04 
for  18  da.  =  18  x  $.000£  =  .003 


The  interest  of  $1  for  2  yr.  8  mo.  18  da.  =  $.163 

The  interest  of  $50.24  is  50.24  times  $.163       =  $8.19 

4.  What  is  the  interest  of  $1  for  2  months?     For  6 
days? 

Rule.  —  Find  the  interest  on  $1  for  the  given  time,  and 
multiply  it  by  the  principal,  considered  as  an  abstract 
number. 

Or,  multiply  the  number  of  dollars  by  the  number  of 
days,  and  divide  by  6.  The  quotient  will  be  the  inter- 
est in  mills. 

Find  the  interest  at  6%  of: 

5.  $382  for  6  mo.  24  da. 

6.  $58.63  for  1  yr.  5  mo.  17  da. 

7.  $256  for  3  yr.  5  mo. 

^  8.  $249.83  for  1  yr.  2  mo.  15  da. 

9.  $51  for  236  da. 

10.  $74  for  2  mo.  19  da. 

11.  $1500  for  1  yr.  3  da. 

12.  $287.15  for  2  yr.  11  mo.  22  da. 

Interest  at  any  rate  per  cent  may  be  found  as  follows  : 
At  7%,  find  interest  at  6%,  and  add  \  of  itself. 
At  5%,  find  interest  at  6%,  and  subtract  \  of  itself. 
At  8%,  find  interest  at  6%,  and  add  f  or  ^  of  itself. 
At  4%,  find  interest  at  6%,  and  subtract  f  or  £  of  itself. 
At  5^%,  find  interest  at  6%,  and  subtract  |  of  itself. 


SIMPLE   INTEREST.  147 

Find  the  interest  and  amount  of  the  following : 

13.  $2350  for  1  yr.  3  mo.  6  da.  at  5%. 

14.  $125.75  for  2  mo.  18  da.  at  7%. 

15.  $950.63  for  3  yr.  17  da.  at  4£%. 

16.  $336.48  for  90  da.  at  1\%. 

17.  $738.53  for  2  yr.  2  mo.  24  da.  at  8%. 

18.  $5000  for  6  mo!  19  da.  at  4%. 

19.  $867.35  for  1  yr.  3  mo.  27  da.  at  9%. 

20.  $260.50  for  5  yr.  21  da.  at  10%. 

21.  $3050  for  3  yr.  3  mo.  3  da.  at  12%. 

22.  $625.57  for  1  yr.  2  mo.  15  da.  at  3%. 

23.  A  grocer's  bill  for  $84.36  is  paid  8  mo.  12  da.  after 
it  becomes  due,  with  interest  at  5%.     How  much  is  paid? 

24.  Find  the  interest  at  7  %  on  $37200  for  5  days. 

25.  A  note  for  $125  was  dated  March  1,  1894.     What 
was  due  Aug.  5,  1895,  int.  at  6%  ? 

26.  Find  the  amount  of  $460.50  for  2  yr.  7  mo.  15  da.  at 
5%. 

27.  What  is  the  amount  of  a  note  for  $360  due  in  3  mo., 
interest  at  5%  ? 

278.  On  short-time  notes,  it  is  customary  to  compute 
interest  for  the  actual  number  of  days,  using  the  6% 
method. 

Find  the  amount  of : 

28.  $684.23  from  June  5, 1895,  to  July  23, 1895,  at  6%. 

29.  $846  from  Jan.  6  to  March  9,  1896,  at  5%. 

30.  $2064.28  from  April  13, 1894,  to  June  3, 1894,  at  8%. 

31.  $1428  from  May  12,  1892,  to  June  9,  1892,  at  6%. 

32.  $324  from  April  1,  1896,  to  June  4,  1896,  at  7%. 


148  SEN lOll    ARITHMETIC. 

33.  $3500  from  Feb.  9, 1895,  to  March  12, 1896,  at 

34.  $862.15  from  May  25, 1893,  to  July  22, 1893,  at  6%. 

35.  What  is  the  amount  of  a  note  of  $384.16  at  6%, 
given  June  11,  1896,  and  paid  Aug.  12,  1896? 

36.  A  note  of  $395.80  dated  April  5,   1896,  was  paid 
Aug.  4,  1896.     What  was  the  amount? 

37.  On  Dec.  9,  1894,  John  Smith  borrowed  $484,  agree- 
ing to  pay  interest  at  5%.     He  paid  the  debt  in  full  on 
March  3,  1895.     What  did  he  pay  ? 

38.  What  is  the  amount  of  $58.24  at  7%  from  April  23, 
1893,  to  July  22,  1893? 

39.  A  bill  of  $312  with  interest  at  5%  was  paid  at  the 
end  of  90  days.     What  was  the  amount  ? 

40.  What  is  the  interest  of  $30000  at  8%  for  7  days  ? 
279.    Find  the  interest,  using  the  best  method. 

PRINCIPAL.  TIME.  RATE. 

41.  $364,  3yr.,  8%. 

42.  $692.15,  1  yr.  3  mo.,  9%. 

43.  $342,  62  da.,  6%. 

44.  $243.50,  2  yr.  5  mo.  18  da.,  7%. 

45.  $.392,  1  yr.  3  mo.  15  da.,  4%. 

46.  $150.16,  7  yr.  2  mo.  27  da.,  4±%. 

47.  $284.10,  1  yr.  8  mo.  18  da.,  $%. 
8.  $1400,  2  yr.  1  mo.  12  da.,  7%. 

49.  $124,  5  yr.  3  ino.  29  da.,  6%. 

50.  $48,  33  da.,  6%. 

51.  $124,  112  da.,  5%. 

52.  $315,  45  da.,  4£%. 

53.  $214,  93  da.,  8%. 


SIMPLE    INTEREST.  149 


280.  Find  the  amount  of : 

54.  $365  from  April  1,  1895,  to  July  5,  1897,  at  6%. 

55.  $250  from  July  3,  1891,  to  April  21,  1893,  at  9%. 

56.  $582  from  Sept.  4,  1896,  to  July  8,  1897,  at  8%. 

57.  $346.18   from   May  10,   1893,   to  March   10,   1895, 
at  6%. 

58.  $287  from  Jan.  1,  1895,  to  July  1,  1897,  at  4J%. 

59.  $1684  from  July  17,  1896,  to  Sept.  5,  1898,  at  7£. 

60.  $2500  from  April  16,  1873,  to  Oct.  11,  1881,  at  5%. 

61.  $186  from  Feb.  12,  1896,  to  March  4,  1896,  at  6%. 

62.  $346  from  March  11,  1895,  to  Feb.  11,  1896,  at  6%. 

EXACT  INTEREST. 

281.  When  the  time  includes  days,  interest  computed  by 
the  6%   method  is  not  strictly  exact,  by  reason  of  using 
only  30  days  for  a  month,  which  makes  the  year  only  360 
days.     The  day  is  therefore  reckoned  as   ^^0  of  a  year, 
whereas  it  is  ^5-  of  a  year. 

Rule.  —  To  compute  exact  interest,  find  the  exact  time  in 
days,  and  consider  1  day's  interest  as  5£g-  of  1  years 
interest.  . 

1.  Find  the  exact  interest  of  $358  for  74  days  at  7%. 

SOLUTION.  —  $358  x  .07  =  $25.06,    1   year's   interest.      74   days' 
interest  is  f&  of  1  year's  interest.     ^  of  §25.06  =  $5.08.     Ans. 

Find  the  exact  interest  of  : 

2.  $324  for  15  d.  at  9%. 

3.  $253  for  98  d.  at  4%. 

4.  $624  for  117  d.  at  7%. 

5.  $153.26  for  256  d.  at  5£%. 

6.  $620  from  Au<j.  15  to  Nov.  12  at  6<&. 


150  SENIOR   ARITHMETIC. 

7.  $540.25  from  June  12  to  Sept.  14  at  8%. 

8.  $7560  for  90  days  at  5^%. 

9.  Find  the  exact  interest  at  5%  on  a  note  dated  Jan. 
14,  1896,  and  paid  March  31,  1896,  for  $832. 

10.  Find  the  exact  interest  on  $800  for  219  days  at  4|%. 

11.  A  city  treasurer  deposits  $387,913.56  in  the  banks 
at  2%  per  annum.     What  interest  will  the  city  receive  in 

5  days  ? 

12.  On  June  4,  1895,  a  coal-dealer  bought  of  the  D.  L. 

6  W.  E.  R.  235  tons  of  chestnut  coal  at  $4.10  per  ton. 
At  6%  what  will  be  the  exact  interest  on  the  amount  on 
Jan.  1,  1896  ? 

PROBLEMS   IN   INTEREST. 

282.    To  find  the  Rate,  when  Principal,  Interest,  and  Time 
are  given. 

1.  What  is  the  rate  when  the  interest  of  $250  for  4 
years  is  $60  ? 

$10/$60  SOLUTION.  —  The    interest   on 

~~6  times  1%  =  6%     the  PrinciPal  at  l?°  for  4  years  = 
$10.     Since  $10  is  the  interest  at 

1  %,  $60  must  be  the  interest  at  as  many  times  1  %  as  $10  is  contained 
times  in  $60,  which  are  6  times.     Therefore  the  rate  is  6  times  \% 

«-e#.' 

Rule.  —  Divide  the  given  interest  by   the  interest  of  the 
principal  for  the  given  time  at  Ify- 

2.  A  man  borrowed  $4625  for  5  yr.  8  mo.  18  da.,  and 
paid  $1586.37^  for  the  use  of  it.     What  was  the  rate  of 
interest  ? 

3.  If  $30.44  is  paid  for  the  use  of  $960  for  7  mo.  18 
da.,  what  is  the  rate  per  cent  ? 

4.  At  what  rate  per  cent  must  $1450  be  loaned  for  4 
yr.  5mo.  to  yield  $576.37  £  ? 


SIMPLE   INTEREST.  151 

5.  At  what  rate  will  $1730  amount  to  $2048.32  in  4 
yr.  7  mo.  6  da.  ? 

6.  4  yr.  7  mo.  6  da.  after  its  date  a  note  for  $1730 
amounted  to  $2048.32.     What  was  the  rate  of  interest  ? 

7.  At  what  rate  %  must  $5600  be  invested  for  1  yr.  4 
mo.  to  bear  $560  interest  ? 

-—8.  A  Kansas  farmer  has  a  mortgage  on  his  farm  for 
$1250.  What  rate  of  interest  does  he  pay,  if  the  interest 
for  2yr.  6mo.  equals  £  of  the  debt  ? 

9.    At  what  rate  must  $2800  be   invested  to  yield  a 
semi-annual  interest  of  $112  ? 

10.  At  what  rate  will  $606  yield  $198  in  5  years  and  6 
months  ? 

11.  At  what  rate  will  any  sum  double  itself  in  20  yr.  ? 

283.  To  find  Time,  when  Principal,  Interest,  and  Rate  are 
given. 

1.  In  what  time  will  $250  gain  $60  at  6%  ? 

SOLUTION.  —  The  interest  of  $250  for  1  year  at  Q%  =  $15.  Since 
$15  is  the  interest  for  1  year,  $60  is  the  interest  for  as  many  years 
as  $15  is  contained  times  in  $60  =  4  years. 

Rule. — Divide  the  given  interest  by  the  interest  of  the 
principal  for  1  year. 

2.  In  what  time   will  $600    yield  $91.50  interest  at 


X  .06  =  $36,  interest  on  principal  for  1  year. 
$91.50  -r  36  =  2.5416+  years.     Reducing  the  decimal  part 
of  the  time  to  months  and  days,  we  have  6  mo.  15  da. 
The  answer  is  2  yr.  6  mo.  15  da. 

NOTE.  —  A  decimal  less  than  .5  of  a  day  is  not  counted,  but  .5  or 
more  is  counted  another  day. 


152  SENIOR    ARITHMETIC. 

3.    In  what   time   will    $530   gain   $92.75  interest   at 


4.  In  what  time  will  $400  yield  $55  interest  at 

5.  In  what  time  will  $500  gain  $15  at  6%  ? 

6.  In  what  time  will  $4625  yield  $1586.38  at  6%  ? 

7.  In  what  time  will   $1730  amount  to  $2048.32   at 
*%? 

8.  The  face  of  a  note  was  $960,  rate  of  interest  5^, 
and  the  interest  $30.44.     How  long  did  it  run  ? 

_—    9.    I  borrowed   $1284   at   4£%,  and    kept   it   until  it 
amounted  to  $1421.067.     How  long  did  I  keep  it? 

10.  For  how  long  will  $2700  have  to  be  invested  to 
amount  to  $2976.25  at  5%  ? 

11.  A  man    received    $9.73   interest   on    $556  at   7%. 
What  was  the  time  ? 

12.  In  what  time  will  any  sum  double  itself  at  6%  ? 

284.  To  find  Principal,  when  Interest  or  Amount,  Rate,  and 
Time  are  given. 

1.  What  principal  at  6%   will  gain  $60  interest  in  4 
years  ? 

$.247  $60.00  (250. 

43  SOLUTION.  —  Since   1   dollar  in  4  years 

:r™  will  gain  $.24  interest,  it  will  take  as  many 

1  OA  dollars  to  gain  $60  interest  as  $.24  is  con- 

tained times  in  $60,  or  $250. 

Rule.  —  Divide  the  given  interest  by  the  interest  of  $1  for 
the  given  time  and  rate. 

2.  What  principal  at  6%   wrill  amount  to  $310  in  4 
years  ? 

SOLUTION.  —  Since  $1.24  is  the  amount  of  $1  for  4  years,  $310 
must  be  the  amount  of  as  many  times  $1  as  $1.24  is  contained  times 
in  310  =  $250. 


SIMPLE   INTEREST.  153 

3.  What  sum  invested  at  5%  will  give  a  yearly  income 
of  $500  ? 

4.  What  principal  will  yield  $25  in  6  mo.  at  5%  ? 

5.  What  principal  in  3  yr.   6   mo.   at  5C/0   will  yield 
$92.75  interest  ? 

6.  What  sum  of  money  will  produce  $1586.37^  in  5 
years,  8  mo.  18  da.  at  6%  ? 

7.  What  principal  will  yield  $318.32  in  4  yr.  7  mo.  6 
da.  at  4%  ? 

8.  What  principal  will  pay  $1556.77^  interest  in  2  yr. 
9  mo.  at  4£%  ? 

9.  The  amount  is  $1093.92^,  time  2  yr.  3  mo.  27  da., 
rate  5  % .     What  is  the  principal  ? 

10.  It  required  $407.65  to  pay  a  loan  at  8%  for  7  mo. 
24  da.  What  sum  was  loaned  ? 

PROMISSORY  NOTES. 

285.  A  Promissory  Note  is  a  written  promise  to  pay  a 
sum  of  money  at  a  certain  time. 

286.  At  least  two  parties  must  be  named  in  the  note, 
the  Maker  and  the  Payee. 

The  Maker  makes  the  promise  to  pay  to  the  Payee  the 
sum  named  in  the  note.  This  sum  is  called  the  Face.  The 
owner  of  a  note  is  called  the  Holder. 

Each  State  has  a  lawful  or  legal  rate  of  interest. 

If  no  rate  is  fixed  in  the  note,  the  legal  rate  is  under- 
stood. 

287.  Interest  higher  than  the  legal  rate  is  Usury. 

288.  A  note  is  Negotiable  when  payable  to  the  bearer,  or 
to  the  order  of  the  payee.     It  is  called  negotiable  because 
it  can  be  negotiated ;  i.e.,  bought  and  sold. 


154  SENIOR    ARITHMETIC. 

289.    The  two  forms  of  notes  given  below  are  negotiable. 

NOTE  1. 

Rochester,  N.  K, 

m&nZTtd  after  date,  &  promise  to  pay  to  the 
rd  r    f  ^T^X  x/   (&/^  l/$f" 

&tve   (SZutndktt/  &n   ant/  5*  Dollars,  for  value    re- 

100 

ceived,  at  the  First  National  Bank. 


NOTE  2. 

Albany,  N.  K,    JU^  /^ 

'  Metii  after  date,  for  value  received,  (§/  promise  to 
pay  (Qy^da  {&.  <Q/U£nei  or  bearer, 
/J_  Dollars,  with  interest. 


NOTE. — A  note  may  be  payable  on  a  given  day  ;  as,  On  March 
15  after  date,  I  promise  to  pay,  etc.  A  note  may  be  payable  on 
demand  ;  as,  On  demand  I  promise  to  pay,  etc. 

290.  A  note  made  payable  to. the  payee  only  is  called 
a  non-negotiable  note. 

NOTE.  — When  a  note  is  payable  in  a  State  in  which  three  days  of 
grace  are  allowed,  maturity  is  three  days  after  the  expiration  of  the 
interval  named  in  the  note. 

291.  1.    Write  a  negotiable  note,  bearing  interest. 
Indorse  it  with  payee's  name,  and  find  the  amount  at 

maturity. 

To  whom  must  the  maker  pay  the  money  ? 


SIMPLE   INTEREST.  155 

2.  "Write  a  non-negotiable  note  payable  on  a  specified 
date,  and  find  the  amount  due  at  maturity. 

3.  Write  a  negotiable  note,  and  find  the  amount  of  it. 

4.  Write  a  non-interest-bearing  demand  note. 

5.  Write  a  non-negotiable  interest-bearing  note. 

6.  Write  a  6%  note,  dated  June  15,  1895,  payable  in  1 
year  without  interest,  with  yourself  as  payee,  and  your 
teacher  as  maker,  and  find  the  amount  of  it  to  the  present 
time. 

7.  Write  a  negotiable  note,  using  the  following  : 
Date,  Jan.  16,  1894  ;  Time,  6  months ;  Face,  $1684.96 ; 

Payee,  Andrew  Jackson ;  Maker,  Silas  Wright ;  Interest 
at  6%.  Indorse  it,  showing  that  the  maker  has  transferred 
it  to  another.  What  is  the  amount  of  the  note,  if  paid  in 
full  Nov.  11,  1894  ? 

292.  A  note  should  contain  : 

1.  The  face  in  figures  at  the  left  upper  corner. 

2.  The  place  and  date  at  the  right  upper  corner. 

3.  The  time  of  payment. 

4.  The  words  "  Value  Received." 

5.  The  face  written  in  words  in  the  body  of  the  note. 

6.  The  place  at  which  it  is  payable. 

7.  The  words  "  with  interest,"  if  agreed  upon. 

293.  A  note  is  said  to  mature  on  the  day  on  which  it  is  due. 

294.  A  note  that  does  not  contain  the  words  "  with  inter- 
est "  bears  interest  from  maturity,  if  not  paid  at  that  time. 

When  does  interest  begin  in  Note  1  ?     In  Note  2  ? 

295.  In  many  of  the  States  the  maker  is  allowed  three 
days  (called  Days  of  Grace)  in  which  to  pay  a  note,  after 
the  time  named  in  the  note  has  expired.     In  these  States, 
the  date  of  maturity  falls  on  the  last  day  of  grace. 

Days  of  grace  are  not  allowed  in  California,  Connecticut, 


156  SEXIOIl    ARITHMETIC. 

District  of  Columbia,  Idaho,  Illinois,  Maryland,  Massachu- 
setts, Montana,  New  Jersey,  New  York,  North  Dakota, 
Ohio,  Oregon,  Pennsylvania,  Utah,  Vermont,  and  Wisconsin. 

NOTE.  — When  the  holder  of  a  note  transfers  it  to  another,  he  is 
usually  required  to  indorse  it,  i.e.,  to  write  his  name  across  the  back. 
This  is  required  as  an  order  to  the  maker  to  pay  the  money,  when 
due,  to  the  new  holder.  An  indorser  is  also  responsible  for  the  pay- 
ment of  a  note  in  case  the  maker  fails  to  pay  it  when  due. 

PARTIAL   PAYMENTS. 

296.  Payments  in  part  of  a  note  or  other  debt  are 
Partial  Payments. 

The  Supreme  Court  of  the  United  States  has  adopted 
the  following  rule  for  finding  the  amount  due  on  a  note 
after  partial  payments  have  been  made. 

UNITED   STATES  RULE. 

Find  the  amount  of  the  principal  to  the  time  when  the  pay- 
ment or  sum  of  the  payments  equals  or  exceeds  the 
interest  then  due. 

Deduct  from  this  amount  the  payment  or  payments. 

Treat  the  remainder  as  a  new  principal,  and  so  proceed 
until  the  date  of  settlement. 

NOTE. — When  a  partial  payment  of  a  note  or  other  contract  is 
made,  the  holder  writes  upon  the  back  of  it  the  sum  paid,  with  the 
date  of  payment.     Sums  so  written  are  called  indorsements.     The 
common  form  of  indorsement  is  as  follows: 
Received  on  the  within, 

July  16,  1896,     $ 

1.  A  note  was  given  Jan.  1,  1892.  and  settled  July  13, 
1894.  The  following  payments  were  indorsed  upon  it : 
May  25,  1892,  $250 ;  Jan.  25,  1893,  $45 ;  April  7,  1893, 
$375;  July  13,  1893,  $750.  How  much  was  due  on  the 
day  of  settlement,  interest  at  6%  ? 

First  write  the  note,  and  properly  indorse  the  payments 
upon  the  back  of  it. 


SIMPLE   INTEREST. 


157 


VR.        WO. 

1892      5 
1892      1 

DA. 

25 

1 

PAYMENTS. 

$250 

$1820  Principal. 
.024 

4 
.024 

24 

§43.68 
1820.00 

$1863.  08  1st  Amount. 
250.00  1st  Payment. 

Ite      1 
1S9X    5/ 

25 
'25 

$45 
$375 

^ijrreiSvGS^New  Principal.^  —  •  —  "•""" 
$>6^J&-hiterest  exceeiis-Ea^ment. 

A° 

>4          \ 

1893      4 
1892      5 

7 
25 

$1613.68 
.052 

$420 

10 
.052 

12 

§583.91 
1613.68 

$1697.59  Amount. 
420.00  Sum  of  2d  and  3d  Payments. 

1893      7 
1893      4 

13 

7 

$750 

$1277.59  New  Principal. 
.016 

3 
.016 

6 

20.44 
1277.59 

$1298.03  Amount. 
750.00  4th  Payment. 

1894      7 
1893      7 

13 
13 

Settled. 

$548.03  New  Principal. 

.06 

1      0 
.06 

0 

$28.88 
548.03 

$576.91     Ans. 

NOTE.  — The  $45  payment,  being  less  than  the  interest  (§64.55), 
is  not  deducted  from  the  amount  of  the  second  principal  ($1678.23). 
If  this  were  done,  and  the  remainder  treated  as  a  new  principal, 
a  portion  of  it  ($19.55),  being  interest,  would  draw  interest,  which 
is  not  legal.  Therefore,  interest  must  be  taken  on  $1613.68  until 
the  date  of  the  next  payment  (10  mo.  12  da.).  The  sum  of  the  two 
payments,  being  greater  than  the  interest,  is  subtracted  from  the 
amount. 

Write  in  proper  form  on  paper  a  note  for  each  of  the 
following,  indorse  the  payments,  and  solve  : 


158  SENIOR   ARITHMETIC. 

2.  Date,  Jan.  1,  1874,  at  Syracuse,  N.Y.     Face,  $1000. 
Interest  at  6%.     Indorsements:  July  7,  1874,  $400;  Oct. 
19,  1874,  $300;   Dec.  1,  1874,  $100.     What  remains  due 
Jan.  1,  1875  ? 

3.  Face,  $900.     Date,  March  1,  1886.     Interest  at  9%. 
Indorsements:  Aug.  10,  1886,  $300;  Sept.  1,  1886,  $100; 
Jan.  1,  1887,  $50.     What  was  due  March  1,  1887? 

4.  Face,  $2000.    Date,  Jan.  20,  1892.    Interest  at  6%. 
Indorsements:  May  20,  1892,  $100;  July  20,  1893,  $100; 
Sept.  10,  1893,  $700;  Oct.  20,  1894,  $75.     Settled  Oct.  20, 
1895.     What  was  due  ? 

5. 

00.  Binghampton,  N.  K, 

On  demand,  for  value  received,  (Q/^  promise  to  pay 

or  order, 
,  with  interest. 


The  following  payments  were  made  on  this  note  :  June 
27,  1891,  one  hundred  fifty  dollars;  Dec.  9,  1892,  one 
hundred  fifty  dollars.  What  was  due  Oct.  9,  1895  ? 

6.  On  a  note  for  $573.25,  at  6%,  dated  June  10,  1888, 
were  the   following   indorsements:    May  20,    1889,   $50; 
July  10,  18'90,  $16.50;  April  5,  1891,  $14.30;  July  14, 

1892,  $250.     How  much  was  due  Sept.  20,  1893? 

7.  A  note  of  $850  was  dated  June  21,  1892,  bearing 
interest  at  6%.     On  this  note  were  the  following  indorse- 
ments: Sept,  15,  1892,  $150.90;  Nov.  21,  1893,  $45;  Jan. 
15,  1894,  $256.88.     What  remained  due  June  21,  1894  ? 

8.  Find  what  was  due  June   1,   1896,  on   a   note  for 
$1928,  with  4%%  interest,  dated  Jan.  1,  1891,  and  bearing 
the  following  indorsements  :  March  1,  1891,  $300;  Oct.  16, 

1893,  $40;  Feb.  4,  1894,  $800;  Dec.  16,  1895,  $500. 


SIMPLE    INTEREST.  159 

9.  On  a  note  for  $832.26  dated  Aug.  3,  1889,  due  in 
6  months,  the  following  payments  were  indorsed :  $350, 
Oct.  5,  1890 ;  and  $468.37,  May  15,  1892.  How  much  was 
due  Dec.  12,  1893,  interest  at  1%  ? 

10.  Face,  $2950.  Date,  July  1,  1885.  Interest,  1%. 
Indorsements :  Oct.  1,  1885,  $750  ;  Jan.  15,  1886,  $600  ; 
July  1,  1886,  $900;  Dec.  1,  1886,  $300;  March  1,  1887, 
$450.  What  was  due  July  1,  1887? 

MERCHANTS'    RULE. 

297.  When  notes  and  accounts  are  settled  within  a  year 
after  interest  begins,  and  upon  which  partial  payments 
have  been  made,  it  is  customary  for  business  men  to  make 
use  of  the  following  rule  : 

Find  the  amount  of  the  entire  debt  at  date  of  settlement. 

Find  the  amount  of  each  payment  at  date  of  settlement. 

Subtract  the  amount  of  the  payments  from  the  amount  of 
the  debt. 


Syracuse,  N.  Y.,  <gMQ0p  /,  18$ 
For  value  received,    ©^  promise  to  pay 

bearer, 
Dollars    on    demand,    with 


Indorsements  :  June  1,  1896,  $150  ;  Aug.  1,  1896,  $200  ; 
Oct.  1,  1896,  $300. 

What  was  due  Dec.  1,  1896  ? 

SOLUTION.  —  $648  in  7  mo.  amounts  to  $670.68 

$150  in  6  mo.  amounts  to  -$154.50 
$200  in  4  mo.  amounts  to  $204.00 
$300  in  2  mo.  amounts  to  $308.00  661.50 

$9.18 


160  SENIOR   ARITHMETIC. 

2.  On  a  note  of  $1186.48,  with  interest  at  5%,  dated 
April  4,  1890,   these   payments  were  indorsed:    July   10, 
1890,  $250;  Aug.  4,  1890,  $300;  Dec.  8,  1890,  $150;  Jan. 
2,  1891,  $75.     How  much  was  due  Feb.  4,  1891? 

3.  On  Oct.  16,  1896,  John  D.  Wilson  gives  his  note  for 
$483.98,  with  interest  at  6%.     He  pays  the  note  in  full, 
March  28^  1897,  having  made  a  payment  of  $350  on  Jan. 
28,  1897.     How  much  does  it  require  to  settle  the  note  ? 

COMPOUND   INTEREST. 

298.  Compound  Interest  is  interest  on  unpaid  interest,  as 
well  as  on  the  principal,  at  the  end  of  regular  interest  periods. 

NOTE.  —  Interest  is  compounded  annually,  semi-annually,  or 
quarterly,  according  to  agreement. 

Compound  interest  is  not  authorized  by  law.  It  is  customary  for 
savings-banks  to  allow  interest  on  interest  when  it  has  been  on 
deposit  for  a  full  interest  period. 

1.  Find  the  compound  interest  of  $350  for  2  years  and 
6  months  at  6%. 

SOLUTION.  —  $350.00  Principal. 

21.00  Interest  for  1st  year. 
$371.00  Amount  taken  as  new  principal. 

22.26  Interest  for  2d  year. 
$393.26  Amount  used  as  new  principal. 
_11-80  Interest  for  6  mo. 
$405.06  Amount  for  2  yr.  6  mo. 
350.00  1st  principal. 
$55.06  Compound  interest  for  2  yr.  6  mo. 

NOTE. — When  the  interest  is  compounded  semi-annually,  the 
rate  is  one-half  the  annual  rate  for  each  period.  When  quarterly, 
one-fourth,  etc. 

When  no  interest  period  is  mentioned,  interest  is  compounded 
annually. 

2.  What  is  the  compound  interest  of  $830  for  3  years 
at  5  per  cent  ? 

3.  What  is  the  amount  of  $650  for  4  years  at  4%  in- 
terest, compounded  semi-annually  ? 


SIMPLE    INTEREST,  1G1 

^^     — • 

4.  What  is  the  compound  interest  of  $365  for  2  yr.  7 
mo.  18  da.  at  6%,  compounded  semi-annually  ? 

5.  What  is  the  compound  interest  on  $640  for  4  years 
at5%? 

6.  What   is   the    interest,    compounded    quarterly,    on 
$538.25  for  2  yr.  6  mo.,  rate  4%  ? 

7.  What    is    the    interest,    compounded    annually,    on 
$683.48  for  4  years  at  6%  ? 

8.  What  is  the  compound  interest  on  $437.50,  3  yr.  6 
mo.,  at  5%,  compounded  semi-annually  ? 

REVIEW   OF   INTEREST 

299.    1.    What  is  simple  interest  ?     Compound  interest  ? 
A  promissory  note  ?     A  negotiable  note  ? 

2.  Define  payee,  holder,  signer  or  maker. 

3.  Describe  two  common  methods  of  computing  interest. 

4.  Prove  that,  at  6%,  6  cents  is  the  interest  on  $1  for  1 
year. 

5.  Prove  that  5  mills  is  the  interest  on  $1  for  1  month. 

6.  Prove  that  £  mill  is  the  interest  on  $1  for  1  day. 

7.  Why  is  interest  not  accurate  when  computed  by  the 
6%  method? 

8.  Find  the  interest  on  $50000  for  252  days  by  the 
6%   method,  then  by  the  exact  interest  method.     Which 
is  more  favorable  to  the  payee  ? 

9.  When  does  a  note  mature  ? 

10.  What  elements  must  be  given  when  we  find  inter- 
est?    Kate?     Time?     Principal? 

11.  How  do  you  find  the  rate  ?     The  time  ?     The  prin- 
cipal ? 

12.  What  are  days  of  grace  ? 


162  SENIOR    ARITHMETIC, 

13.  Does  the  maker  of  a  non-interest-bearing  note  ever 
have  to  pay  interest  ?     Explain. 

14.  What  use  is  made  of  compound  interest  ? 

15.  Find  the  compound  interest,  then  the  simple  interest, 
at  6%  on  $25000  for  5  years,  and  note  the  difference. 

16.  When  a  note  is  not  paid  at  maturity,  why  is  it  to  the 
holder's  advantage  to  require  a  new  note  ? 

17.  What  is  the  effect  of  a  payee's  indorsement  ? 

18.  When  a  partial  payment  is  made  that  does  not  equal 
the  interest  due,  why  is  not  the  payment  subtracted  from 
the  amount  ? 

19.  Solve  a  problem  in  partial   payments  by  both  the 
United  States,  and  the  merchants'  rule.     Which  is  more 
favorable  to  the  payer? 

20.  Find  the  compound  interest  on  $1420.80  for  1  yr. 
9  rno.  at  6%,  computed  semi-annually. 

21.  Find  the  amount  of  a  debt  of  $5672.00  for  4  years 
at  4%  compound  interest. 

22.  Find  the  interest  on  $720  at  6%   for  2  yr.  8  mo. 
22  days. 

Find  the  interest  on  : 

23.  $675.20  for  3  pr.  5  mo.  at  7%. 

24.  $754.30  for  1  yr.  4  rno.  15  da.  at  5J%. 

25.  $564.11  for  2  yr.  3  mo.  18  da.  at  4%. 

—  26.    A  county  in  Missouri  owes  $85.640.     In  how  many 
days  will  the  interest  at  6%  amount  to  $897.22  ? 

27.  Find  the  interest  at  8%  on  $3960.36  for  9  mo.  20 
days. 

28.  Find  the  amount  of  $2536.48  for  1  yr,  3  mo.  18  da. 
at  7%. 

29.  The  interest  on  $600  for  3  yr.  6   mo.  was  $126. 
What  was  the  rate? 


SIMPLE    INTEREST.  163 

30.    The   interest  on  a  note   for    $460.50   at  5%    was 
).44.     What  was  the  time  ? 
_.31.    The  interest  on  a  certain  sum  was  $96.04,  the  rate 
6%.     Find  the  principal. 

32.  The  amount  due  on  a  6%  note  due  in  1  yr.  5  mo. 
I  da.  was  $135.708.     What  was  the  face  of  the  note  ? 

33.  Find  the  exact  interest  on  a  note  for  $600,  dated 
Aug.  5,  1895,  and  due  July  1,  1896,  interest  at  6%. 

34.  Find  the  amount  and  simple  interest  of  $623.74,  one 
half  of  which  is  to  be  paid  in  2  yr.  3  mo.  at  4%,  the  other 
half  to  be  paid  in  3  yr.  5  mo.  at  6%. 

35.  A  note  for  $146.20,  dated  June  5,  1869,  was  paid 
July  11,  1872,  with  interest  at  6  per  cent.     What  was  the 
interest  ? 

36.  A  man  borrowed,   Dec.   25,   1877,   $137.40   at  6% 
interest,  and  kept  it  until  Jan.  15,  1880.    What  was  the 
interest  ? 

37.  Payments  were  made  on  a  note  of  $1800  dated  Jan. 
12,  1891,  as  follows:  March  6,  1891,  $300;  April  15, 1891, 
$190;  July  3,  1891,  $565;  Oct.  15,  1891,  $700.     What 
was  due  Dec.  21,  1891,  interest  at  6%  ? 

38.  When  must  $1600  be  put  at  interest  at  6%,  so  that 
it  will  amount  to  $1800  on  Jan.  1,  1898  ? 

39.  Find  the  amount  of  $375  for  2  yr.  8  mo.  16  da.  at 
6%. 

40.  Find  the  amount  at  simple  interest  of  $1200  from 
April  4,  1895,  to  the  present  time. 

41.  A  note  for  $728  is  dated  Nov.  16,  1894.     March  8, 
1895,  there  was  paid  on  it  $25.     Find  the  amount  due  on 
Jan.  4,  1896,  interest  at  6%. 

42.  Find  the  amount  at  simple  interest  of  $1184.63  for 
1  yr.  4  mo.  17  da.  at 


164  SENIOR   ARITHMETIC. 

43.  Write    your   own   promissory  note   for  $200,    with 
interest,  payable  in  60  days  from  to-day.     When  does  it 
become  due?     Find  the  amount  due  at  maturity. 

44.  Find  the  exact  interest  on  $843.20  from  April  10, 
1895,  to  March  15,  1896,  at  4J%. 

45.  Upon  a  note  for  $950,  dated  Syracuse,  N.  Y.,  Jan. 
1,  1894,  $150  was  paid  Aug.  16,  1894 ;  $25  March  1, 1896 ; 
and  $200  April  16,  1896.     How  much  is  due  to-day  ? 

46.  $645  was  paid  as  interest  on  $2000  for  3  yr.  7  mo. 
What  was  the  rate  ? 

47.  $30  was  paid   as  interest  on  $600  at  6%.     What 
was  the  time  ? 

_48.    A  house  that  cost  $5000  was  rented  for  $500,  and 

$100  was  paid  for  annual  taxes  and  repairs.     What  rate  of 
interest  did  the  investment  yield  ? 

49.  A  person  investing  a  certain  sum  of  money  at  6% 
for  1  yr.  6  mo.  found  at  the  end  of  that  time  the  invest- 
ment amounted  to  $545.     Find  the  sum  invested. 

50.  A  man  bought  a  horse  for  $150,  paying  $70  in  cash, 
and  the  balance  on  time  at  6%.     He  paid  at  the  time  of 
settlement  $83.60.     How  much  time  elapsed  before  that 
date  ? 

51.  H.  C.  Harmon  loaned  $250  for  1  yr.  3  mo.  27  da., 
which    amounted   to  $269.875   at   the  time  of   payment. 
Find  the  rate  of  interest. 

52.  A  person  having  a  certain  sum  of  money  invested, 
and  drawing  compound  interest  at  6%,  found  at  the  end  of 
2  yr.  2  mo.  that  it  amounted  to  $567.418.     WThat  was  the 
sum  invested  ? 

53.  A  sum  of  money  was  borrowed  Jan.  30,  1895,  and 
$419.60  paid  in  full  Nov.  24,  1895.      The  rate  of  interest 
being  6%,  how  much  of  this  was  interest  ? 


TRUE   DISCOUNT.  165 

54.  A  man  owes  $4600  at  7%,  and  each  payment  of 
interest  amounts  to  $161.  How  often  does  he  pay  in- 
terest ? 

TRUE   DISCOUNT. 

300.  Oral. 

1.  What  will  be  the  amount  of  $100  at  6%  one  year 
from  to-day  ? 

2.  What  is  the  value  to-day  of  a  debt  of  $106,  due  in 
one  year,  when  money  is  worth  6%  interest  ? 

3.  How  much  money  paid  to-day  will  cancel  a  debt  of 
$112,  due  two  years  hence,  money  being  worth  6%  ? 

4.  What  is  the  present  worth  of  $105,  due  in  one  year 
without  interest,  when  money  is  worth  5%  interest  ? 

5.  W7hen  money  can  be  loaned  at  7%,  which  is  worth 
the  more,  $100  at  the  present  time,   or  a  note  of   $107 
without  interest,  due  in  one  year  ? 

6.  What  sum  should  be  deducted  from  a  debt  of  $108, 
due  without  interest  in   one  year  in  consideration  of   its 
being  paid  now,  when  money  can  be  loaned  at  8%  ? 

301.  True  Discount  is  a  deduction    of   interest  for  the 
payment  of  a  debt  before  due. 

302.  The  Present  Worth  of  a  debt  due  at  a  future  time 
is  a  sum  which  will  amount  to  the  debt  if  put  at  interest 
till  that  time. 

The  debt  is  therefore  the  amount  of  the  present  worth 
for  the  given  time. 

303.  The  true  discount   is  the    difference   between   the 
debt  and   its  present  worth.     It   is    the   interest   of   the 
present  worth  for  the  given  time. 

7.  What  is  the  present  worth  and  the  true  discount  of 
a  debt  of  $582.40,  due  in  8  mouths  without  interest,  when 
money  is  worth  6^  ? 


166  SENIOR    ARITHMETIC. 

SOLUTION.  —  $582.40  -f  $1.04  =  $560,  present  worth. 
$582.40  -  $560  =  $22.40,  true  discount. 

Since  $1.04  is  the  amount  of  $1  for  8  mo.,  $582.40  is  the  amount 
of  as  many  dollars  as  $1.04  is  contained  times  in  $582.40  =  $560. 

Rule.  —  To  find  the  present  worth,  divide  the  debt  by  the 

amount  of  $1  for  the  given  time. 

To  find  the  true  discount,  subtract  the  present  worth  from 
the  debt. 

8.  What  is   the   present  worth   and  true  discount  of 
$400,  due  in  one  year,  when  money  is  worth  5%  ? 

9.  A  father  wills  his  two  sons  $3000  each,  to  be  paid 
in  three  years  from  the  time  of  his  death.     What  is  the 
value  of  the  legacies  at  the  probate  of  the  will,  if  money 
is  worth  6%  ? 

10.  What  is  the  present  worth  of   $450,  due  in   two 
years  at  5%  ? 

11.  What  is  the  present  worth  of   $250.51  payable  in 
8  months,  money  being  worth  6<fa  ? 

12.  Which  is  better,  to   buy  flour  for   $5  cash,  or  for 
$5.25  on  6  months'  time,  when  money  can  be  borrowed  at 
5%? 

13.  Find  the  present  worth  of  $750  for  6  months,  money 
being  worth  6%. 

14.  What  is  the  present  worth  of  $600,  due  in  1  year 
without  interest  ? 

15.  Write  the  note  which  would  be  given  for  the  above 
debt. 

16.  A  man  wishing  to  buy  a  house  and  lot  has  his  choice 
between  paying  $5400  in  cash,  or  $4000  in  cash  and  $1700 
in  two  years.     With  money  at  6%,  which  is  the  most  ad- 
vantageous for  him  ? 


BANK    DISCOUNT.  167 

17.  Which  would  be  more  profitable,  and  how  much,  to 
pay  $4000  cash  for  a  house,  or  $4374.93  in  3  yr.  6mo., 
money  being  worth  7%  ? 

18.  I  can  sell  my  house  for  $2800  cash,  or  $3000  and 
wait  6  months  without  interest.     I  choose  the  latter  ;  do  I 
gain  or  lose,  and  how  much,  money  being  worth  6%  ? 

19.  What  is  the  present  worth  of  a  debt  of  $385.31,  due 
in  5  months,  15  days,  at  6%  ? 

BANK  DISCOUNT. 

304.  When  the  holder  of  a  negotiable  note  wishes  the 
money  before  it  becomes  due,  he  may  take  it  to  a  commer- 
cial bank  ;  and  if  the  banker  is  satisfied  that  the  parties  to 
the  note  are  responsible,  he  will  pay  the  holder  the  amount 
due  after  deducting  the  discount.     By  this  act  the  bank 
becomes  the  holder  of  the  note,  and  at  its  maturity  the 
maker  must  pay  to  the  bank  instead  of  to  the  payee. 

305.  The  Maturity  Value  of  a  note  is  the  amount  due  at 
maturity. 

The  Bank  Discount  is  the  simple  interest  on  the  maturity 
value,  reckoned  from  the  day  of  discount  to  the  day  of 
maturity. 

306.  The  maturity  value  less  the  bank  discount  is  called 
Proceeds,  or  Avails.     The  time  from  the  day  of  discount  to 
the  day  of  maturity  is  called  the  Term  of  Discount. 

307.  The  maturity  value  of  a  note,  not  bearing  interest 
is  the  face,  and  the  maturity  value  of  an  interest-bearing 
note  is  the  face  plus  the  interest. 


1.  —  Only  short-time  notes  are  discounted  at  banks,  usually 
not  exceeding  4  months. 

NOTE  2.  —  Banks  generally  require  that  the  paper  which  they 
discount  be  made  payable  at  some  bank. 

NOTE  3.  —  Banks  usually  reckon  discount  for  the  exact  number  of 


168  SENIOR    ARITHMETIC. 

days  in  the  term  of  discount,  although  the  time  in  a  note  may  be 
expressed  in  months.  Banks  usually  regard  the  year  as  360  days. 

NOTE  4.  —  In  States  having  days  of  grace,  the  day  of  maturity  is 
the  last  day  of  grace. 

NOTE  5.  —  If  the  day  of  maturity  falls  on  Sunday  or  a  legal  holi- 
day, the  preceding  day  is  the  day  of  maturity  in  most  States.  In  some 
States,  however,  the  note  does  not  fall  due  until  the  day  following. 

NOTE  6.  —  When  a  note  is  discounted  at  date,  the  term  of  dis- 
count is  the  time  of  the  note  (4-3  days  of  grace  in  States  having 
days  of  grace). 

Unless  otherwise  stated,  a  note  is  to  be  discounted  at  date. 


/33<5  ,w  Syracuse,  N.  K,    J 


after  date,  for  value  received,  (Qs  promise 
/g&* 

x?hO/;  x?  A 

<£//ii€€  wtnt/ied  fatenrftf/t'ltt  tint/  ^  Dollars,  at  the 
Third  National  Bank. 


If  this  note  was  discounted  at  6%  at  a  bank  on  the  day 
it  was  made,  how  much  did  the  bank  deduct  ?  How  much 
were  the  proceeds  ? 

SOLUTION.  —  The  term  of  discount  is  from  Jan.  15  to  Mar.  15  = 

JAN.  FEU.  MAR. 

16  da.  +  29  da.  +  15  da.  -  60  da. 

The  bank  discount  is  the  interest  of  $325 rVo  for  60  da.,  at 
6^  =  $3.25;  the  proceeds  =  $325.24  -  $3.25  =  $321.99. 

2.  Copy  the  above  note,  and  properly  indorse  the  payee's 
name  across  the  back. 

3.  What  would  be  the  bank  discount  and  proceeds  of  the 
above  note  if  it  contained  the  words  "  with  interest." 


BANK    DISCOUNT.  169 

SOLUTION. — 

Maturity  value  =  face  +  interest,  or  $325.24  +  $3.25  =  $328.49. 
The  bank  discount  =  Q%  of  $328.49  for  60  da.  =  $3.28. 
The  proceeds  =  $328.49  -  $3.28  =  $325.21. 

4. 

Boston,  Mass.,  ^tme 

/?  ^-J] 

m&nt/td  after  date,  &  promise  to  pay  to  the 

/&/         /        /     /          /x  / 

%3'sUtt  fwnme-a  &mw<fu  deven  ana  ^-  Dollars,  value 

received,  at  the  First  National  Bank,  with  interest  at  <5% . 


TOO' 


Discounted  July  27  at  5%. 
Day  of  maturity,  Sept.  27. 

SOLUTION.  — 

Maturity  value  =  face  +  interest  for  90  da.,  at  5%  =  $392.34. 
Discount  at  5%  from  July  27  to  Sept.  27,  62  da.  =  $3.38. 
Proceeds  $392.34  -  $3.38  =  $388.96. 


Buffalo,  N.Y.,tin.  3-/, 
<m#n<w  after  date,  ffl  promise  to  pay  to  the 
order  of^^^^^^^^i^me 

&fe?  rtitndt^  &0ife-<eta4M  &n^  Dollars,  at  the  Shoe 

v       </      # 

and  Leather  Bank,  value  received^  with  interest  at  6°jo, 

Discounted  at  date  at  6%. 

The  above  note  is  interest-bearing,  therefore  the  discount 
must  be  computed  on  the  amount  at  maturity. 


170  SENIOR    ARITHMETIC. 

6. 

^3000.  Detroit,  Mich.,  $0S  St  1896 

(S/ftnt-frM  tfrezud  after  date,  for  value  received,  ©t  promise 

to  pay '/Cw.-\^.-x^         ^-J^1/-^^^^  ^&s!o     ^^I'CvK^n^  or  order 

*/,  at  the  First  National  Bank. 


Discounted  at  date  at  5. 


7. 


/43&3f.  St.  Louis,  Mo.,  &ed  f,  1893. 

(S/4&&  m&nwtd  after  date,  for  value  received,  (S/  promise 

</  (/  IOO 

Chemical  National  Bank. 

Discounted  March  10  at  6%. 


/oo  Cleveland,  O.,  ^ 


•m&n-Md  after  date,  (Qt  promise  to  pay  to  the  order 

^^yg'ltgn  ™>tf?lfM'g(/  €t'il'M/fo-ntn€-  -tinu    ~—  Dollars,  Value 
received,  at  the  City  Bank.  (f^.^'' 

Discounted  May  4  at  6%. 


BANK    DISCOUNT.  171 


to  pay  to  the  order  of^ 


Brooklyn,  N.  Y.,  &W4.  3,  189$. 
after  date,  for  value  received,  (Qr  promise 


ne  weeMdfoet/  -and  Stvtn^u  Dollars, 
at  the  Merchant  'j  Bank,  with  interest. 


Discounted  at  date  at  6%, 

10. 

JVJ/J.  Boston,  Mass.,  IL//7?-tl<u  -/,  /< 

after  date,  for  value  received,  ^^  promise 

/^y          /s^/fr  *?))//••/ 
to  Pay  to  the  order  of  ^-^^^..  \UutZ(ll  (bx/.     vv/^^Z^^^v^ 

j:    s  ^  ,/ 

at  the  First  National  Bank. 
Discounted  May  10  at  4%. 


Albany,  N.  Y., 

after  date,  for  value  received,  (QP  promise 

^or  order, 
$L  Dollars, 

ith  interest,  at  the  Park  Bank.  g&. 

Discounted  Feb.  27,  1897,  at  6%. 


172  SENIOR    ARITHMETIC. 

12. 

0  X 

San  Francisco,  CaL,    J><ttM£  </0,  189®. 

£>*^  r. 

after  date,  for  value  received,  &/  promise 

'•Wl&tti  *$£f.     ^/W^iZ1  or  order 

^ 

frttdant/ Dollar s,  at  the  Citizen 's  Bank. 


Discounted  July  10  at  8%. 

13. 

Syracuse,  JV.  Y , 
after  date,  ©^ promise  to  pay  to  the  order 


ollars,  at  the  Third 
National  Bank.  J^K  'Q. 

Discounted  at  6%  at  date. 

14.  A  note  for  $135  is  given  for  90  days,  and  discounted 
the  day  it  is  given  at  6%.  What  are  the  proceeds  ? 

15. '  William  Johnson  gave  John  Doe  a  note  payable  to 
the  Binghamton  Trust  Co.,  time  60  days,  amount  f  204.60. 
Write  this  note.  After  20  days  Doe  put  the  note  in  the 
bank.  What  are  the  proceeds  of  the  note  ? 

In  the  following  problems,  write  the  notes  in  full,  and 
properly  indorse  them,  using  any  names  for  payer  and 
payee. 

16.  Find  the  bank  discount  of  $400  for  3  months  at  8%. 

17.  What  are  the  proceeds  of  $250,  with  interest  at  6%, 
discounted  at  bank  for  60  days  at  6%  ? 


DISCOUNT.  173 

18.  What  will  be  the  proceeds  of  a  note  for  $175  drawn 
at  4  mo.,  with  interest  at  6%.  if  the  bank  discount  is  10% 
per  annum  ? 

19.  On  the  first  day  of  January,  1896,  a  farmer  gave  his 
note  at  90  da.  for  $525,  with  interest  at  6%.     When  did 
the  note  become  due?  and  what  were  the  proceeds  of  the 
note  if  discounted  at  a  bank  at  1%  a  month  on  the  tenth 
day  of  February  ? 

308.  To  find  the  Face  of  a  note,  when  the  Proceeds,  Time, 
and  Rate  are  known. 

1.  What  must  be  the  face  of  a  60-day  note,  without 
grace,  which  after  being  discounted  at  6%  will  give  $500 
as  proceeds  ? 

SOLUTION.  — 

The  bank  discount  of  $1  at  $%  for  60  da.  =  $.01. 
The  proceeds  of  $1  =  $1.00  -  $.01  =  $.99. 

Since  $.99  is  the  proceeds  of  $1,  $500  must  be  the  proceeds  of  as 
many  dollars  as  $.99  is  contained  times  in  $500  —  505.05+  Ans. 

Therefore,  $505.05  must  be  the  face  of  a  60-day  note  which  will 
give  $500  as  proceeds  after  being  discounted  at 


Rule.  —  Divide  the  proceeds  by  the  proceeds  of  $1. 

2.  A  person   must  use   $250   to-day.     For   how  much 
must  he  make  a  bank  note  for  three  months  that  will  give 
$250  proceeds,  without  grace  ? 

3.  What  must  be  the  face  of  a  60-day  note,  payable  at  a 
Boston  bank,  upon  which  I  can  realize  $350  after  it  is 
discounted  at  6%  ? 

4.  If  you  buy  goods  for  $1200  cash,  how  large  a  note 
payable  in  90  days,  at  6%  bank  discount,  must  you  make 
that  the  proceeds  shall  pay  for  the  goods  ?    Without  grace. 

5.  Find  the  face  of  a  60-day  note  that  will  yield  $800 
when  discounted  at  bank  at  7%,  with  grace. 


174  SENIOR    ARITHMETIC. 

6.  How  large  a  note  must  I  make  at  a  bank  for  30  days 
to  pay  a  debt  of  $475,  without  grace  ? 

7.  Wishing  to  borrow  $ 494.90  at  a  Syracuse  bank,  for 
what  must  I  make  my  note  at  60  da.,  with  interest  at  6%, 
in  order  to  obtain  this  amount  ?     Discount  at  1  %  a  month. 

8.  The  proceeds  of  a  Buffalo  note  at  60  da.,  when  dis- 
counted at  a  bank  at  6%  per  annum,  is  $742.50.     What  is 
the  face  ? 

REVIEW   OF   DISCOUNT. 

309.  1.  Define  Discount ;  True  Discount ;  Bank  Dis- 
count ;  Proceeds ;  Present  Worth. 

2.  How  is  the   present  worth  found  ?     The   true  dis- 
count ?     The  bank  discount  ?     The  proceeds  ? 

3.  What  is  the  term  of  discount  ?  and  how  is  it  found  ? 
The  day  of  discount  ?  and  how  found  ? 

4.  How  is  the  bank  discount  of  an  interest-bearing  note 
found  ? 

5.  How  do  you  find  the  face  of  a  note  when  the  pro- 
ceeds, time,  and  rate  are  given  ? 

6.  When  does  a  "Rochester,  N.Y.,  note  mature  if  given 
for  1  month  from  Jan.  31  ? 

7.  State  a  point  of  difference  between  true  discount  and 
bank  discount. 

8.  What  kind  of  notes  only  can  be  discounted  at  banks  ? 
^ — 9.    Bought  a  city  lot,  and  agreed  to  pay  $546.94  at  the 
end  of  2  yr.  6  mo.,  without  interest.    Receiving  some  money 
unexpectedly  after  6  months,  I  wish  to  pay  cash.     How 
much  ought  I  to  pay,  money  being  worth  6%  ? 

10.  What  is  the  present  value  and  true  discount  of 
$973.52,  due  in  1  yr.  7  mo.  24  da.  hence,  without  interest, 
money  being  worth  8%  ? 


DISCOUNT.  175 

11.  A  man  has  an  offer  of  $2000  cash  for  his  house,  or 
$2100  payable  in  8  months.     If  money  is  worth  8%,  which 
is  the  better  ?  and  how  much  ? 

12.  Find  the  discount  and  proceeds  of  a  note  for  $13500 
payable  at  a  bank  in  90  days  after  date  without  grace,  dis- 
counted at  5%. 

13.  For  what  sum  must  C.  F.  Norton  draw  his  note  on  a 
Binghamton  bank,  that  when  it  is  discounted  at  4%  for  60 
days  he  will  have  $800  ? 

14.  A  man  owes  me  $2540,  due  in  2  years,  3  months,  with- 
out interest.     If  he  pays  it  at  once,  what  discount  should  I 
allow  him  ? 

15.  Find  the  discount  and  proceeds  of  a  note  on  a  Brook- 
lyn bank  for  $350,  given  May  12,  1896,  for  4  months,  and 
discounted  at  6%,  July  15. 

16. 

Syracuse,  N.  Y., 

%4  after  date,  for  value  received, 

order, 
Dollars,  with  interest,  at  the  Salt 

( 

Springs  National  Bank. 

Discounted  June  11  at  6%. 

17.  For  what  sum  must  I  draw  a  4  months'  note  so  that 
the  proceeds  will  be   $800,   discounted   without  grace  at 
6%? 

18.  I  sell  my  horse  for  $216,  and  take  a  note  due  in  6 
months  without  interest.     If  money  is  worth  6^  per  an- 
num, what  is  the  present  value  of  my  note  ? 


176  SENIOR    ARITHMETIC. 

19.  For  what  sum  must  I  give  my  note  for  60  days  at 
a  bank  in  order  to  receive  $650  proceeds,  money  being 
worth  8%  ? 

20.  Find  the  face  of  a  note,  discounted  for  $2558.40  at 
8%,  for  a  term  of  72  days,  without  grace. 


STOCKS  AND  BONDS. 

310.  Many  kinds  of  business  require  so  much  capital 
that  several   persons  must  unite  to   raise   the    necessary 
amount. 

311.  The   Capital  to   be  raised  is  divided  into  Shares, 
usually  of  $100  each. 

Shares  are  then  sold  until  the  required  amount  is  raised. 

Each  purchaser  of  shares  is  a  Stockholder,  and  receives 
a  Certificate  of  Stock,  which  shows  the  number  of  shares 
purchased,  and  their  value. 

This  value  is  called  the  Par  or  Nominal  Value. 

312.  The  Market  Value  of  stocks  is  the  price  for  which 
they  are  sold. 

313.  The  value  of  stocks  depends  upon  the  profitable- 
ness of  the  business.    When  the  business  is  very  profitable, 
the  shares  are  worth  more  than  par;  they  are  then  abov^- 
par,  or  at  a  Premium. 

When  the  business  is  unprofitable,  the  shares  are  not 
worth  their  par  value.  They  are  then  below  par,  or  at  a 
Discount. 

314.  1.    The  capital  of   a   company  is  $100000.     Into 
how  many  shares  of  $100  each  can  this  be  divided  ? 

2.    A  stockholder  owns  25  shares  of  stock.     How  many 
dollars  of  stock  has  he  ? 


STOCKS   AND   BONDS.  Ill 

3.    If  at  the  end  of  a  year  there  has  been  a  net  profit 
of  $  10,000,  what  per  cent  profit  has  been  made  ? 

$10000  is  what  %  of  $100000  ? 

The  profits  are  divided  among  the  stockholders,  and  are 
called  Dividends. 


1.  —  Dividends    are    usually    declared    semi-annually    or 
quarterly. 

NOTE  2.  —  When  a  10  %  dividend  is  declared,  each  stockholder 
receives  10%  of  the  par  value  of  his  shares. 

4.  What  will  be  A's  dividend  if  he  owns  35  shares  ? 
When  there  is  a  loss,  each  stockholder   is   required  to 

pay  his  share  of  the  loss.     This  is  called  an  Assessment. 

5.  What  would  be  A's  assessment  to  meet  a  2%  loss  ? 
A  person  who  buys  or  sells  stocks  for  others  is  called  a 

Stock-broker,  and  his  commission  is  called  Brokerage. 

NOTE    1.  —  Brokerage  is  usually  \  %  or  |  %  of  the  par  value. 

NOTE  2.  —  In  all  stock  transactions,  dividends,  assessments, 
brokerage,  premium,  and  discount  are  computed  on  the  par  value. 

NOTE  3.  —  Shares  are  sometimes  issued  at  $200,  $250,  $50,  $25, 
or  $10  each,  but  unless  otherwise  stated  $100  is  considered  the  par 
value  of  a  share. 

6.  What  is  the  market  value  of   10    shares    of   bank 
stock,  when  sold  at  par  ? 

7.  What  is  the  market  value  of  50  shares  of  railroad 
stock,  at  10%  premium  ? 

SOLUTION.  —  The  market  value  of  1  share  is  $100  +  $10  =  $110. 
The  market  value  of  50  shares  is  50  times  8110. 

8.  What  is  the  market  value  of  18  shares  of  mining 
stock  at  15%  below  par? 

SOLUTION.  —  The  value  of  1  share  is  $100  —  §15  =  $85. 
The  value  of  18  shares  is  18  times  $85. 


178  SENIOR    ARITHMETIC. 

Stock  Quotations  are  the  published  prices  of  stocks. 
When  railroad  stock  is  quoted  at  108,  it  means  that  it 
sells  for  8%  above  par  in  the  stock-market. 

When  it  is  quoted  at  92,  it  is  selling  at  8^  below  par. 

9.    If  I  buy  stock  at  98  and  sell  it  at  101,  what  gain  do 
I  make  on  10  shares  ? 

NOTE.  —  Stock  at  98  means  $98  for  a  $100  share,  and  stock  at  101 
means  $101  for  a  $100  share. 

10.  When  stock  is  quoted  at  85,  what  is  the  value  of  a 
share  ?     What  is  the  value  of  1  dollar  of  stock  ? 

11.  What  must  I  pay  for  10  shares  of  stock  at  95,  if  I 
pay  the  broker  \c/0  for  doing  the  business  ? 

SOLUTION.  —  Cost   of    1    share  =  $95  +  Brokerage  $i  =  $95i  or 

$95.25. 
$95.25  x  10  =  $952. 50.     Ans. 

12.  If  I  sell  10   shares  of  stock   at  110,  and  pay  the 
broker  \°/oi  what  do  I  receive? 

SOLUTION.  —  1  share  brings  $110  —  $|  =  $109f ,  or  $109.75. 

10  shares  bring  10  times  $109.75  =  $1097.50.     Ans. 

13.  A  man  invested  $4500  in  street  railway  stock    at 
10%  discount.     How  many  shares  did  he  purchase? 

14.  If  I  invest  $2100  in  bank  stock  at  105,  ho\v  many 
shares  do  I  purchase  ? 

15.  A  capitalist  bought  80  shares  railroad  stock  at  87^, 
and  60  shares  mining  stock  at  114.     Find  the  cost. 

16.  $18200   will    purchase    how   many   shares  of    stock 
selling  at  140  ? 

17.  A  stock  company  declares  a  dividend  of  2£%.    What 
does  A  receive,  who  owns  1500  shares  of  $10  each  ? 

18.  How  much  is  gained  on  50  shares  of  railroad  stock 
purchased  at  98  and  sold  at  102  ? 


STOCKS    AM)    BONDS.  179 

19.  Bought  stock  at  a  discount  of  2%,  and  sold  it  at  a 
discount  of  3  % .  Did  I  gain  or  lose  ?  and  how  much  on 
20  shares? 

BONDS. 

315.  To    meet    extraordinary    expenses,    governments, 
States,  cities,  villages,   counties,   towns,   and  incorporated 
companies  sometimes  borrow  money.     The  securities  given 
by  such  corporations  are  called  Bonds. 

Bonds  bear  a  fixed  rate  of  interest,  payable  annually, 
semi-annually,  or  quarterly.  They  are  bought  and  sold  in 
the  same  manner  as  stocks. 

Bonds  are  known  by  the  rate  of  interest  they  bear :  Vir- 
ginia 6's  are  bonds  of  the  State  of  Virginia,  bearing  6%  ; 
U.  S.  4's  of  '97  are  U.  S.  bonds  bearing  4%  interest,  and 
maturing  in  1897. 

316.  A   Coupon  is   an  interest  certificate  attached  to  a 
bond.     At  the  expiration  of  any  interest  period,  the  coupon ' 
is  cut  off  and  used  in  collecting  the  interest,  being  worth 
the  amount  of  interest  due  on  the   bond   for  a  specified 
period. 

317.  20.    What  will  be  the  cost,  including  brokerage  at 
1%,  of  200  shares  of  C.,  B.,  and  Q.  R.R.  bought  at  67|  ? 

SOLUTION/.  —  Cost  of  1  share  =  $67|  +  $i  =  $68J.  Cost  of  200 
shares  =  200  x  $68  J. 

21.  How  much,  including  brokerage  at  |%,  must  be  paid 
for  $5000  of  U.  S.  4's  at  110J  ? 

SOLUTION.  —  $1  of  bonds  costs  $1.10|  +  .OOJ  =  $1.11.  $5000 
worth  will  cost  5000  times  $1.11. 

22.  What  must  I  pay  for  $8275  of  stock  at  10%  dis- 
count ? 

23.  What  is  the  cost,  including  broker's  commission  of 
%%,  of  150  shares  of  railroad  stock  bought  at  89J  ? 


180  SENIOR    ARITHMETIC. 

24.  I  buy  stocks  at  5%  discount,  and  sell  at  5%  pre- 
^nium  ;  what  per  cent  profit  do  I  make  on  the  investment  ? 

25.  March  10,  1896,  Western  Union  Telegraph  stock  was 
quoted  at  84£.     How  many   shares  could   be  bought  for 
$1020,  brokerage  |  per  cent  ? 

SOLUTION.  —  Cost  of  1  share,  $84|  +  g  =  $85.     As  many  shares 
can  be  purchased  as  $85  is  contained  in  $1020. 


26.  How  many  shares  of  stock  at  10%  premium  can  be 
purchased  for  $2200  ? 

27.  I  invested  $5100  in  K.Y.  and  N.H.  railroad  stock  at 
170.     How  many  shares  did  I  purchase  ? 

28.  If  I  invest  $42400  in  5%  bonds  at  106,  what  is  my 
yearly  income  ? 

SOLUTION.  —  $42400  -f-  $1.06  =  $40000,  par  value.     How  much  is 
5^  of  $40000? 

29.  If  I  invest  $21008  in  5%  bonds  at  104,  what  will  be 
my  annual  income  ? 

30.  What  will  be  my  yearly  income  if  I  invest  $11100 
in  5%  at  92,  brokerage  \%  ? 

31.  A  man  invests  $9500  in  Virginia  6's  at  94|,  broker- 
age i  <f0  .     What  is  his  quarterly  income  ? 

32.  What  will  be  my  annual  income  if  I  invest  $5050  in 
4  cfo  water  bonds  at  1  %  premium  ? 

33.  What  is  my  dividend  on  80  shares  of  electric-light 
stock,  when  a  5  °/o  dividend  has  been  declared  ? 

34.  What  sum  must  be  invested  in  Chicago  5's  at  92  to 
yield  an  income  rof  $600,  brokerage  1%  ? 

SOLUTION.  —  $600  -f  .05  =  $12000,  par  value.     How  much  is  92£  % 
of  $12000  ? 

35.  How  much  must  I  invest  in  4%  bonds  at  8%   pre- 
mium, to  secure  an  annual  income  of  $200  ? 


STOCKS   AND   BONDS.  181 

36.  How  much  must  be  invested  in  city  3^'s  at  8%  dis- 
count, to  secure  an  income  *of  $350  ? 

37.  How  much  telegraph  stock  must  1  sell  at  ll£%  Dis- 
count, brokerage  J%,  to  realize  $8800  ? 

38.  I  invested  through  a  broker  $5450  in  stock  at  1.08£, 
brokerage  %%.     How  much  did  I  purchase  ? 

39.  I  sell  through  a  broker  enough  stock  at  4|  c/c  premium 
to  realize  $10475,  brokerage  \°/o-     How  much  do  I  sell  ? 

40.  What  rate  of  interest  do  I  receive  on  my  investment 
if  I  buy  7%  stock  at  112  ? 

SOLUTION.  —Each  share  of  stock  costs  $112,  and  yields  $7  inter- 
est.    $7  is  what  per  cent  of  $112  ? 

41.  Stock  yielding  7%  annually  is  bought  at  111|.    What 
annual  rate  of  income  will  it  yield  on  the  investment  ? 

42.  What  rate  will  6%  bonds  pay  on  the  investment  if 
bought  at  112  ? 

43.  What  is  the  rate  on  Syracuse  4's  at  a  premium  of 


44.  What  is  the  rate  of  income  on  6's  at  90,  no  brokerage  ? 

45.  Which  is  the  better  investment,  6's  at  par,  or  5's  at 
a  discount  of  12^%  ? 

46.  How  much  must  I  pay  for  5%  stock  to  secure  annu- 
ally 7%  on  my  investment  ? 

SOLUTION.  —  1  share  of  5%  stock  yields  $5  interest  annually;  this 
$5  is  1%  of  the  cost  of  1  share.  Therefore  the  question  is,  $5  is  1% 
of  what? 

47.  At  what  price  must  5%  stock  be  purchased  so  that 
it  will  yield  4%  on  the  investment  ? 

48.  How  much  must  I  pay  for  5's  to  make  my  invest- 
ment yield  6%  ? 

49.  What  must  I  pay  for  city  6's  that  my  investment 
may  yield  8%  annually? 


182  SENIOR    ARITHMETIC. 

50.  How  much  must  I  pay  for  1  share  of  3%  stock,  that 
the  dividend  may  be  4  %  of  the  purchase  price  ? 

51.  How  much  will  be  my  income  if  I  invest  $2300  in 
4%  bonds  at  115? 

SOLUTION.  —2300  -j-  $1.15  =  $2000  par  value.     How  much  is  ±% 
of  2000  ? 

52.  What  sum  invested  in  Tennessee  6's  at  85  will  yield 
an  annual  income  of  $1800  ? 

53.  How  much  money  must  I  invest  in  6%  stock  at  80 
to  secure  an  annual  income  of  $3186  ? 

54.  I  want  an  income  of  $1500.     How  much  shall  1  in- 
vest in  5%  stocks  at  25%  premium  to  secure  that  amount  ? 

55.  How  much  must  a  man  invest  in  a  5%  stock  at  120 
to  yield  him  an  annual  income  of  $2500  ? 

MISCELLANEOUS. 

318.      1.    At  what  premium  should  an  8%  stock  sell  to 
yield  a  6  %  income  ? 

2.  A  man  bought  stock  at  3^%  discount  and  sold  it  at 
2%    premium,  paying  a  brokerage  of  \  %    in  both  cases. 
His  net  profit  was  $680.    How  much  money  did  he  invest? 

3.  A  man  invested  his  money  in  6%   railroad  stocks, 
and  received  $300  semi-annually.     What  was  the  sum  in- 
vested ? 

*   4.    Which  is  the  better  investment,  and  how  much,  a 
4%  stock  bought  at  85,  or  a  6%  stock  bought  at  120  ? 

5.  What   rate   on   the   investment  do   7%    stocks   pay 
when  bought  at  a  premium  of  8%  ? 

6.  What  sum  must  be  invested  in  U.  S.  6%  bonds  to 
yield  an  income  of  $1000  ? 

7.  What  sum  must  be  invested  in  U.  S.  6's  at  92^  per 
share  to  yield  a  quarterly  dividend  of  $300  ? 


AVERAGE   OF    PAYMENTS.  183 

8.  At  what  price  should  8%  bonds  be  bought  to  make 
the  income  from  the  investment  equivalent  to  that  from 
6%  bonds  at  par? 

9.  Which  is  the  better  investment,  4%  bonds  at  86,  or 
6%  bonds  at  105? 

10.  How  much  must  I  pay  for  a  4^  stock  that  the  in- 
vestment may  yield  me  6%  V     For  a  1  c/c  stock  that  the 
investment  will  yield  oc/o  ? 

11.  If  25  shares  of  stock  paying  8%  are  sold  at  150,  and 
the  proceeds  loaned  at  6%,  will  the  income  be  increased  or 
diminished  ?  and  how  much  ? 

12.  Bought  bonds  at  125  and  sold  them  at  110,  thereby 
losing  $600.     How  many  $1000  bonds  did  I 'buy  ? 

13.  How  many  dollars  of  stock  can  I  buy  for  $105000 
if  stock  is  quoted  at  120  ?    How  many  shares  ?    What  per 
cent  do  I  receive  on  my  investment  if  the  stock  bears  6%  ? 

14.  What  is  the  cost  of  200  shares  of  D.,  L.,  &  W.  K.R. 
at  162^  ?    If  it  pays  a  quarterly  dividend  of  2%,  what  is 
the  yearly  income  from  this  investment  ?     What  rate  does 
it  pay  on  the  investment  ? 

15.  B  invests  $1680  in  a  stock  selling  at  112.     What 
does  he  receive  from  a  dividend  of  4%  ? 

16.  An  estate  derives  an  annual  income  of  $3600  from 
stock  that  pays  7|-%.     How  many  $25  shares    does  the 
estate  own  ? 

AVERAGE    OF  PAYMENTS. 

319.      1.    The  use  of  $5  for  2  mo.  equals  the  use  of  $1 
for  how  many  months  ? 

2.    The  use  of  $10  for  6  mo.  will  balance  the  use  of  $5 
for  how  many  months  ? 

SOLUTION.  —  The  use  of  $10  for  6  mo.  =  the  use  of  $1  for  60  mo. 
The  use  of  $1  for  60  mo.  =  the  use  of  $5  for  £  of  60 
mo.  =  12  mo. 


184  SENIOR    ARITHMETIC. 

3.  How  long  may  $20  be  kept  to  balance  the  use  of  $5 
for  20  months  ?     $50  for  10  mo.  ? 

4.  A  credit  of  $10  for  8  mo.  equals  a  credit  of  $20 
for  how  many  months  ? 

5.  The  interest  of  $500  for  1  year  equals  the  interest 
of  $100  for  how  long  ?     Prove  this. 

6.  I  pay  a  debt  of  $20  four  months  before  it  is  due. 
How  long  after  it  is  due  should  my  creditor  allow  a  debt 
of  $40  to  remain  unpaid  ? 

A  person  owing  two  debts  due  at  different  times  may 
pay  both  at  an  intermediate  time  without  loss  to  himself 
or  his  creditor,  by  paying  one  of  them  before  it  is  due  and 
the  other  an  equivalent  time  after  it  is  due. 

320.  The  process  of  finding  the  time  when  several  debts 
due  at  different  times  can  be  equitably  discharged  at  one 
payment  is  called  Average  of  Payments. 

321.  The  date  of  such   payment  is  called  the  Average 
Time,  and  the  time  to  elapse  before  the  payment  is  made  is 
called  the  Average  Term  of  Credit. 

NOTE.  —  The  time  to  elapse  before  any  debt  becomes  due  is  called 
a  Term  of  Credit. 

322.  When  the  terms  of  credit  begin  at  the  same  date. 

1.    On  Jan.  8,  A  bought  goods  on  the  following  condi- 

tions  :  $300  due  in  2  months. 

$200  due  in  4  months. 

$100  due  in  6  months. 

How  long  after  Jan.  8  may  the  debt  be  equitably  dis- 
charged at  one  payment  ? 

SOLUTION.  — 

A  credit  of  $300  for  2  mo.  .-=  a  credit  of  $1  for    600  mo. 

A  credit  of    200  for  4  mo.  =  a  credit  of  $1  for    800  mo. 

A  credit  of    100  for  6  mo.  =  a  credit  of  $1  for    600  mo. 

A  is  entitled  to  a  credit  of  $1  for  2000  mo. 


AVERAGE    OF    PAYMENTS.  185 

A  credit  of  $1  for  2000  mo.  =  a  credit  of  $600  for  -b4TT  of  2000 
mo.,  or  3^  mo.  =  3  mo.  10  da.,  average  term  of  credit. 
Jan.  8  +  3  mo.  10  d.  =  April  18,  equated  time.     Ana. 

Short  method. 

2  mo.  x  300  =    600  mo. 

4  mo.  x  200  =    800  mo. 

6  mo.  x  100  =    600  mo. 

600    /  2000  mo. 

3£  mo.  =  3  mo.  10  da. 
Jan.  8  +  3  mo.  10  da.  =  April  18. 

NOTE.  —  One-half  a  day  or  more  is  called  another  day.    Less  than 
^  day,  not  counted. 

Call  50^  or  more  $1.00.     Less  than  SO'',  not  counted. 

Rule. — Multiply  each  debt  by  its  term  of  credit.  The 
sum  of  the  products  divided  by  the  sum  of  the  debts 
irill  be  the  average  term  of  credit. 

2.  Gates  Thalheimer  sold  a  bill  of  goods  on  the   fol- 
lowing terms  :  $325  due  in  60  days,  $175  due  in  90  days, 
and  $185  due  in  4  months.     What  is  the  average  term 
of  credit?  and  on  what  day  may  the  entire  debt  be  paid 
without  loss  to  either  party  ? 

3.  A    merchant    bought    $1000    worth    of    goods,    and 
agreed  to  pay  for  them  as  follows  :  $100  cash  ;  $300  in 
3  mo. ;   $250  in  4  mo.  ;  and  the  balance  in  5  mo.     In  what 
time  could  he  equitably  pay  the  entire  amount  ? 

4.  On  the  first  day  of  April,  1895,  a  man  gave  3  notes, 
one  for  $250  due  in  30  da.,  one  for  $375  due  in  40  da., 
and  one  for  $425   due  in  60  da.     What    is    the    average 
term  of  payment  ?  and  when  could  they  have  all  been  paid 
at  once  ? 

5.  D.  McCarthy  &  Co.  sold  goods  amounting  to  $4000, 
payable  as  follows  :   |  in  three  months,  \  in  4  months,  and 
the  balance  in  5  months.     What  was  the  average  term  of 
credit  ? 


186  SENIOR    ARITHMETIC. 

6.  A  merchant  sold  goods  on  the  following  terms :  1 
payable  in  2^  months,  \  in  3i  months,  1  in  5^  months,  and 
the  balance  in  6  months.     What  was  the  average  term  of 
credit  ? 

7.  Equate  the  following  payments:  $580.75  due  in  30 
days,  $650.25  due  in  60  days,  $450.36  due  in  90  days,  and 
$600  due  in  5  months. 

8.  On  the  1st  of  May  a  merchant  bought  goods  amount- 
ing to  $1500,  agreeing  to  pay  for  them  as  follows  :  $521.35 
on  the  10th  of.  June,  $398.84  on  the  16th  of  July,  $199.60 
on  the  15th  of  August,  and  the  balance  on  the  1st  of  Sep- 
tember.    Upon  what  date  can  he  pay  the  whole  amount  ? 

9.  Jacob  Amos  sold  a  bill  of  flour  amounting  to  $2500, 
payable  as  follows:  $500  due  in  4  months,  $600  due  in 
5  months,  and  the  balance  due  in   6  months.     What  was 
the  equated  time  ? 

10.  A  purchased  a  farm  for  $3000,  agreeing  to  pay  for 
it  as  follows :  $500  cash,  $600  in  5  months,  $1000  in  8 
months,  and  $400  in  1  year.  He  decides  to  give  a  note 
for  the  whole  amount.  When  was  the  balance  to  be  paid  ? 

323.    When  the  terms  of  credit  begin  at  different  dates. 
1.    A  purchased  goods  of  Dey  Bros.  &  Co.,  as  follows: 

Jan.  8,  1895.  $200  on  2  months'  credit. 
Feb.  16,  1895.  $400  on  3  months'  credit. 
April  4,  1895.  $300  on  4  months'  credit. 

Find  the  average  time. 

NOTE.  —  First  find  the  date  when  each  item  is  due. 

$200  due  Mar.   8.     200 

400  due  May  16.     400  x     69  da.  =  27600 

300  due  Aug.   4.     300  x  149  da.  =  44700 

900  72300 


AVERAGE   OF    PAYMENTS.  187 


72300  +  900  =  80^  da.  =  80  da. 

March  8  +  80  da.  =  June  27,  average  time. 

The  first  debt  is  due  March  8,  and  the  last  Aug.  4.  The  average 
time,  therefore,  will  be  between  these  dates. 

$200  due  March  8  has  no  longer  time  to  run. 
$400  due  May  16  has  69  days  after  March  8. 
$300  due  Aug.  4  has  149  days  after  March  8. 

A  is  therefore  entitled  to  a  credit  of  $1  for  72300  da.  after  March 
8,  which  is  equal  to  a  credit  of  $900  for  80  da.  after  March  8. 

Rule.  —  find  the  date  on  which  each  debt  becomes  due,  and 

using  the  earliest  of  these   as  a  standard  date,  reckon 

the  time  to  each  of  the  others. 
Multiply  each  debt  by  its  time,  and  divide  the  sum  of  the 

products  by  the  sum  of  the  debts. 
The  quotient  will  be  the  average  term  of  credit,  which  add 

to  the  standard  date  to  find  the  average  time. 

2.  Four  notes  are  due  as  follows  :  March  4,  $165 ;  April 
15,  $325.50;  May  9,  $94;  June    6,  $465.     What  is  the 
average  date  of  payment  ? 

3.  A  retail  dealer  bought  the  following  bills  of  goods 
on  4  months'  credit:  April  4,   $480;   April  26,  $185.65; 
June  1,  $480.16  ;  July  6,  $196.     What  is  the  average  time 
for  payment  ? 

4.  Bought  goods  as  follows :  Jan.  1,  $250  at  3  mo. ; 
Feb.  1,  $500  at  4  mq. ;  March  11,  $106  at  60  da.     What 
is  the  average  date  of  payment  ? 

5.  Mr.  B  owes  $1000,  due  in  5  months ;  in  2  months  he 
pays  $600.     How  long  after  the  expiration  of  the  5  months 
should  the  remainder  be  paid  ? 

SOLUTION.  —  $600  has  been  paid  3  months  before  due,  which 
equals  a  credit  of  $1  for  1800  months.  He  is  entitled  to  a  like 
credit  for  $400  after  it  is  due.  yfo,  of  1800  mo.  =  4£  months.  Ans, 


188  SENIOR    ARITHMETIC. 

6.  A  lady  purchased  a  piano   for  $500  on  6  months7 
credit.     If  she  pays  $200  cash,  how  long  after  the  expira- 
tion of  the  6  months  should  the  balance  be  allowed  to  run  ? 

7.  May  1,  1896,  a  man  buys  a  store  and  fixtures  for 
$2650,  giving  his  note  payable  in  6  months  without  inter- 
est.    June  15,  he  pays  $500 ;  Aug.  1,  $750.    When  should 
the  balance  be  paid  ? 

8.  G.  L.  Hoyt  purchased  goods  of  Mann  &  Hunter  to 
the  amount  of  $3000  :    $1200  to  be  paid  June  2,  1896  ; 
$600  to  be  paid  July  5,  1896 ;  $200  to  be  paid  Aug.  15, 
1896.     The  balance  will  become  due  Aug.  30,  1896.     At 
what  date  must  a  note  payable  in  3  mo.  be  drawn  that  it 

may  become  due  at  the  average  date  ? 

• 

QUESTIONS. 

324.    1.    Define  discount ;  present  worth ;  true  discount. 
Tell  how  to  find  present  worth  and  true  discount. 

2.  Define  bank  discount ;   proceeds ;  day  of  maturity ; 
term  of  discount. 

Tell  how  to  find  bank  discount  and  proceeds. 
Tell  how  to  find  face  of  note  when  proceeds,  time,  and 
rate  are  given. 

3.  What    is    a    stock    company  ?      What    are    stocks  ? 
Bonds  ?     Shares  ? 

4.  Define  par  value  ;  market  value". 

5.  What  is  a  stock  certificate  ? 

6.  1  >efine  dividend  ;  assessment. 

7.  Upon  what  are  premium,  brokerage,  dividends,  and 
assessments  reckoned  ? 

8.  What  is  the  average  of  payments  ?     Equated  time  ? 
Average  term  of  credit  ? 


RATIO   AND    PROPORTION.  189 

RATIO  AND   PROPORTION. 

325.  Oral. 

1.  5  bears  what  relation  to  10  ?     Ans.    5  is  ^  of  10. 

2.  10  bears  what  relation  to  5  ?     Ans.    10  is  2  times  5. 

3.  What  part  of  16  is  4  ? 

4.  How  does  $7  compare  with  $14  ? 

5.  John  has  20/  and  Mary  5/.     What  is  the  relation 
of  John's  money  to  Mary's  ?     Of  Mary's  money  to  John's  ? 

6.  What  is  the  relation  of  15  to  3  ?     Of  $8  to  $16  ? 
Of  28  men  to  7  men  ?     Of  2  bushels  to  2  pecks  ? 

326.  Ratio  is  the  relation  between  two  like  numbers.     It 
is  found  by  dividing  one  by  the  other ;  thus  : 

The  ratio  of  4  to  8  is  4  -=-  8  =  f 

The  sign  of  ratio  is  (:).  It  is  the  division  sign  with  the 
line  omitted. 

The  ratio  of  6  to  3  is  expressed  thus,  6:3.  It  may  also 
be  expressed  fractionally,  thus,  §. 

327.  The  Terms  of  a  ratio  are  the  two  numbers  com- 
pared. 

The  first  term  of  a  ratio  is  the  Antecedent,  and  the 
second  the  Consequent. 

In  the  ratio  6  : 12,  6  is  the  antecedent,  and  12  the  conse- 
quent. 

328.  A  ratio  formed  by  dividing  the  consequent  by  the 
antecedent  is  an  Inverse  ratio. 

12  -=-  6  is  the  inverse  ratio  of  6  :  12. 

329.  The  two  terms  of  a  ratio  taken  together  form  a 
Couplet. 

330.  Two  or  more  couplets  taken  together  form  a  Com- 
pound ratio. 


190  SENIOR  ARITHMETIC. 

3  :  6  "l  A  compound  ratio  may  be  changed  to  a  sim- 
o  .  K  I Qfi  •  1  50  ple  ratio  ^  takin£  tne  product  of  the  antece- 

T  ~  dents  for  a  new  antecedent,  and  the  product  of 

4  :  5  J  the  consequent  for  a  new  consequent. 


Antecedent  -=-  Consequent  =  Ratio. 

Therefore,  Ratio  -f  Antecedent  =  Consequent; 

and,  Ratio  x  Consequent  =  Antecedent. 

Multiplying  or  dividing  both  terms  of  a  ratio  by  the  same  number 
does  not  change  the  ratio. 

The  ratio  12  :  6  =  2. 

The  ratio  3x12:3x6  =  2. 

The  ratio  12  -f  3  :  6  -f  3  =  2. 

Find  the  ratio  of : 

7.  56  :  7.  11.    3  bu.  :  3  pk.  14.    2^  :  16. 

8.  20  :  300.  12.    J  :  4.  15.    1 :  §. 

9.  $55:  $330.         13.    12  :  J.  16.    3f  :  5f 
10.    What  is  the  ratio  of  T9n  to  ^  ? 

NOTE.  —  Fractions  with  a  common  denominator  have  the  same 
ratio  as  their  numerators.  Prove  this  in  Ex.  10,  by  multiplying  both 
terms  by  10. 

17.  A:i»  =  ?     ff:775  =  ?     H'«=? 

18.  |  :  f  =  ?      f  :  !  =  ?     i  :  f  =  ?     §  .  5  =  ? 

19.  Find  the  inverse  ratio  of  75  to  25.     Of  15  to  225. 

20.  16  :  (?)  =  £.      14  :  (?)  =  2. 

21.  (?):5  =  4.      (?):8  =  J. 

22.  Find  the  value  of  the  compound  ratio,  8  : 10 ") 

5:6    ) 

PROPORTION. 

331.    Oral. 

23.  Give  three  fractions  having  the  same  value  as  §. 

24.  Give  two  numbers  that  have  the  same  ratio  as  5  to  10. 

25.  Give  a  fraction  equal  to  f . 


RATIO   AND   PROPORTION.  31 

26.  Give  a  ratio  equal  to  3  :  4. 

27.  How  does  the  ratio  of  5  men  to  10  men  compare 
with  the  ratio  of  $ 5  to  f  10  ? 

28.  How  does  the  ratio  of  8  Ib.  to  4  Ib.  compare  with 
the  ratio  of  40/  to  20/? 

29.  Name  two  numbers  that  have  the  same  relation  as 
5  to  10.      As  4  to  24.     As  8  to  16.     As  J-  to  J. 

30.  What  number  has  the  same  relation  to  5  as  12  to  3  ? 

31.  Find  a  number  whose  ratio  to  4  equals  3  :  6. 

32.  Give  three  ratios  equal  to  $  100  :  $50. 

33.  Give   any   two    ratios    that   equal    each   other,  and 
express  their  equality. 

332.  An  equality  of    ratios  is  a  Proportion.     Thus,  4: 
2  =  12  :  6.     The  ratio  of  4  to  2  equals  the  ratio  of  12  to  6. 

A  proportion  is  usually  expressed  with  the  sign  (::) 
between  the  ratios  ;  thus,  4  :  2  : :  12  :  6.  This  is  read  4  is 
to  2  as  12  is  to  6. 

A  proportion  has  four  terms,  of  which  two  are  antece- 
dents and  two  are  consequents.  Each  term  is  a  propor- 
tional. 

333.  The  first  and  fourth  terms  are  called  Extremes,  and 
the  second  and  third  terms  are  called  Means. 

NOTE.  — In  the  proportion  2:6  ::  6  : 18,  the  two  means  are  the 
same  number,  6.  The  6  is  called  a  mean  proportional. 

PRINCIPLE.  —  The  groduct  of  the  extremes  equals  the 
product  of  the  means. 

Rule.  —  To  find  an  extreme,  divide  the  product  of  the  means 

by  the  given  extreme. 

To  find  a  mean,  divide  the  product  of  the  extremes  by  the 
given  mean. 


192  SENIOR   ARITHMETIC. 

Supply  the  missing  term  : 

34.  1:836::25:(      ).         39.  K)  yd.  :  50  yd.  ::  $20  :  (     ). 

35.  6  :  24  ::  (     )  :  40.  40.  15  Ib.  :  60  lb.  ::  ($  )  :  $12. 

36.  (     )  :  15  : :  60  :  6.        41.  1  da. :  (  da.)  ::  12  :  6. 

37.  25  :(     )  ::  4  :  8.  42.  (     men)  :  75  ::  $50  :  $150. 

38.  6:4::  J:  (     ).  43.  $ f  :  $3f  ::  (     )  :  5. 

SIMPLE    PROPORTION. 

334.  An  equality  of  two  simple  ratios  is  a  Simple  Pro- 
portion. 

It  is  employed  in  solving  questions  having  three  given 
terms,  two  of  which  have  the  same  relation  to  each  other 
as  the  third  to  the  required  term. 

44.    If  12  bushels  of  oats  cost  $4,what  will  60  bushels  cost  ? 

SOLUTION.  —  There  must  be  the  same  relation  between  the  coxt 

of  12  bu.  and  the  cost  of  60  bu.  as  exists  between  12  bu.  and  60  bu. 

We  place  $4  for  the  third  term.      The 

12  :  60  : :  $4  :  ($      )     answer  wn]  ^e  the  fourth.     We  must  now 

,.~         .  form  a  ratio  of  12  and  60  that  shall  equal 

-  =  $20.  the  ratio  of  $4  to  the  answer.     Since  the 

third  term  is  less  than  the  required  answer, 

the  first  must  be  less  than  the  second,  and  we  have  12  :  60  for  the 
first  ratio.  The  product  of  the  means  divided  by  the  given  extreme 
will  give  the  other  extreme,  or  $20.  Ans. 

By  analysis,  —  Since  12  bu.  cost  $4, 

1  bu.  will  cost  $J,  and 
60  bu.  will  cost  $20.     Ans. 

Rule.  —  Consider  the  required  answer  as  the  fourth  term, 

and  place  the  number  that  is  like  it  for  the  third  term. 
Place  the  two  remaining  terms  as  follows  : 
If  the  answer  is  to   be  larger  than  the  third  term,  the 

second  must  be  larger  than  the  first.     If  smaller,  the 

second  must  be  smaller  than  the  first. 
Divide  the  product  of  the  means  by  the  given  extreme. 

Cancel  when  possible. 


11ATIO   AND    PROPORTION.  198 

45.  If   10   sheep  cost  $35,   what   will   23    sheep   cost? 
What  will  6  sheep  cost  ? 

46.  If  5  men  can  do  a  piece  of  work  in  9  days,  how  long 
will  it  take  15  men  to  do  the  same  work  ? 

SOLUTIOX.  —  Place  9  days  for  the  third  term,  because  it  is  like 
the  required  answer,  thus, 

::9  da.  :  (    da.) 

Since  5  men  can  do  it  in  9  days,  15  men  can  do  it  in  less  time. 
Therefore,  since  the  answer  is  to  be  smaller  than  the  third  term, 
place  5  men  for  the  second,  and  15  men  for  the  first.  Multiplying 
and  dividing  we  have  3  days,  Ans. 

f — 47.    If  14  horses  eat  36  tons  of  hay  in  a  certain  time,  how 
many  tons  will  13  horses  eat  in  the  same  time  ? 

48.  If  it  costs  $400  to  lay  80  rods  of  street-car  track, 
how  much  will  it  cost  to  lay  3^  miles  at  the  same  rate  ? 

49.  If  a  pole  8  ft.  high  casts  a  shadow  4^  ft.  long,  how 
high  is  a  tree  which  casts  a  shadow  48  ft.  long? 

50.  If  a  man  walks  280  miles  in  8  days,  how  many  days 
ought  it  to  take  him  to  walk  420  miles  ? 

51.  If  it  costs  $13.20  to  supply  a  new  arithmetic  to  each 
of  a  class  of  24  pupils,  what  will  be  the  expense  of  furnish- 
ing one  to  each  of  a  class  of  19  ? 

52.  How  far  can  a  train  run  in  3  hours,  if  it  can  run 
160  Km.  in  4  hours? 

53.  How  many  men  will  be  required  to  do  in  10  days 
what  15  men  can  do  in  30  days  ? 

54.  What  will  8  tons  of  coal  cost,  when  171  tons  cost 
$78.75? 

55.  If  a  certain  sum  of  money  yields  $360  interest  in 
one  year,  what  would  the  interest  of  the  same  sum  be  for 
15  months  ? 


194  SENIOR    ARITHMETIC. 

56.  If  $800  yield  $48  interest  in  a   certain  time,  how 
large  a  sum  will  yield  $216  in  the  same  time  ? 

57.  If  the  interest  of  $3600  for  a  certain  time  is  $216, 
what  will  be  the  interest  of  $800  for  the  same  time  ? 

,~-  58.  If  a  garrison  of  240  soldiers  have  a  supply  of  food 
sufficient  for  150  days,  how  long  would  the  same  food  last 
if  the  garrison  were  increased  to  600  men  ? 

59.  In  the  above  example,  how  long  would  the  food  last 
if  80  men  were  sent  away  ? 

60.  Find  the  cost  of  l|f  bushels  of  wheat,  if  f  bu.  costs 


61.  If  120  shoemakers  make  40  dozen  pair  of  shoes  in 
a   certain    time,  how  many  shoemakers   would   it  require 
to  make   the   same    number   of    shoes   in  one-half  of  the 
time? 

62.  If  a  train  runs  140  miles  in  4  hr.  30  min.,  what  is 
the  rate  per  hour  ? 

63.  When  5  tons  1650  Ib.  of  coal  cost  $24.75,  what  will 
be  the  cost  of  18  tons  675  Ib.  ? 

64.  If  a  16-foot  board  9  inches  wide  contains  12  sq.  ft., 
how  wide  must  a  board  of  the  same  length  be  to  contain 
20  sq.  ft.  ? 

65.  It  takes  26  yards  of  carpet  1  yard  wide  to  cover  a 
floor.     How  many  yards  will  it  take  if  the  carpet  is  but  27 
inches  wide  ? 

66.  The  ratio  of  Simon's  pay  to  Matthew's  is  f  .     Simon 
earns  $18  per  week  ;  what  does  Matthew  earn  ? 

67.  25  men  can  do  a  piece  of  work  in  70  days  ;  but  after 
30  days  15  of  them  refuse  to  work.     In  how  many  days 
can  the  rest  complete  the  work  ? 


RATIO   AND   PROPORTION.  195 


COMPOUND    PROPORTION. 


335.  An  equality  between  a  compound  and  a  simple 
ratio  is  a  Compound  Proportion;  thus, 

O    .     A         \ 

>  : :  12  :  20  is  a  compound  proportion, 
o  :  10  ) 

Find  the  fourth  term. 

SOLUTION.  —  First  changing  to  a  simple  pro- 
3:6)  portion,  we  have, 

4:g|::3:(      )  3x4:6x8::3:(). 

Then  divide  the  product  of  the  means  by  the 
given  extreme,  using  cancellation.    Thus, 

gxgxg-M.     An*. 
3  x  * 

1.  If  5  men  earn  $72  in  8  days,  how  much  can  10  men 
earn  in  6  days  ? 

SOLUTION.  —  Since  the  answer  is  to  be  in  dollars,  place  $72  for 

the  third  term,  and  arrange  the 

5  men  :  10  men )        «  terms  of  each  couplet  accord- 

8  days  :  6  days    (  '  ing  as  the    answer  should   be 

greater  or  less  than  the  third 
term  if  it  depended  on  that  couplet  alone. 

Since  5  men  earn  $72,  10  men  can  earn  more,  so  we  place  10  men 
for  the  second  and  5  men  for  the  first ;  and  since  they  earn  §72  in  8 
days,  they  will  earn  less  in  6  days,  so  we  place  6  days  for  the  second 
term,  and  8  days  for  the  first.  Dividing  the  product  of  the  means 
by  the  extremes,  we  have, 

9        2 
&v<i  w  in  ^  « 

Ans. 


By  analysis. 

Since, .        5  men  in  8  days  earn  $72 ; 
Therefore,  1  man  in  8  days  will  earn  $^2-; 
1  man  in  1  day  will  earn  $f ; 

10  men  in  1  day  will  earn  $-%° ; 

10  men  in  6  days  will  earn  ($108). 


196  SENIOR    ARITHMETIC. 

Rule.  —  Consider  the  answer  as  the  fourth  term,  and  place 

the  number  that  is  like  it  for  the  third. 
Arrange  the  couplets  as  if  the  answer  depended  on  each 

couplet  alone,  as  in  simple  proportion. 
Divide  the  product  of  the  means  by  the  product  of  the 

extremes. 
Cancel  when  possible. 

2.  If  4  horses  eat  10  bushels  of  oats  in  5  days,  how 
many   bushels  will   be   required   to    feed  5   horses    for   4 
days  ? 

3.  If  10  men  working  8  hours  a  day  can  do  a  piece  of 
work  in   12  days,  how  many  days  would  it  take  6  men, 
working  10  hours  a  day,  to  do  the  same  amount  of  work  ? 

4.  If  a  wheelman  rides  144  miles  in  3  days  of  6  hours 
each,  how  many  miles  can  he  ride  in  5  days  of  9  hours 
each  ? 

5.  A  section  of  a  street  33  feet  long  and  20  feet  wide 
can  be  paved  with  15840  stones,  each  9  inches  long  and  8 
inches  wide.     How   many  stones   12   inches  long  and  10 
inches  wide  will  it  take  to  pave  a  street  12  rods  long  and 
16  feet  wide  ? 

6.  If  it  costs  $84  to  carpet  a  room  24  feet  long  and 

21  feet  wide  with  carpet  27  inches  wide,  how  much  will  it 
cost  to  carpet  a  room  25  feet  long  and  18  feet  wide  with 
carpet  1  yard  wide  ? 

7.  If  18  men  chop  360  cords  of  wood  in  12  days  of  9 
hours  each,  how  many  cords  could  17  men  chop  in  13  days 
of  10  hours  each  ? 

8.  If  50  men,  working  10  hours  a  day  for  11  days,  can 
dig  25  rods  of  a  canal  60  ft.  wide,  5  ft.  deep,  how  many 
rods  of  a  canal  90  ft.  wide,  7  ft.  deep,  can  140  men  dig  in 

22  days  of  8  hours  each  ? 


RATIO    AND    PROPORTION.  197 

9.    If  60  men  can  build  a  wall  150  ft.  long,  64  ft.  high, 

2  ft.  thick,  in  8  days  of  10  hours  each,  how  many  days  of 
8  hours  each  will  36  men  require  to  build  a  wall  180  ft. 
long,  80  ft.  high,  2£  ft.  thick  ? 

10.  How  many  men  will  it  require  to  mow  48  acres  in 

3  days  of  12  hours  each,  if  6  men  mow  24  acres  in  4  days 
of  9  hours  ? 

11.  If  4  Ib.  6  oz.  of  tea  cost  $2^,  what  will  3  Ib.  11  oz. 
cost  at  same  rate  ? 

. 12.    If  sufficient  flour  to  fill   8   bags  containing  98  Ib. 

each  can  be  produced  from  16  bushels  of  wheat,  how  many 
bushels  will  be  needed  to  fill  14  barrels  of  196  Ib.  each  ? 

13.  My  gas  bill  for  the  month  of   November  is  $3.50 
when  I  use  6  burners  3£  hours  each  evening.     How  much 
ought  it  be   for  the  month  of  December,  when   I  use  4 
burners  for  5  hours  each  evening? 

14.  How  long  a  piece  of  cloth  .4  m.  wide,  can  be  made 
from  175  Kg.  of  wool,  if  45  Kg.  make  a  piece  25  m.  long 
and  .6  m.  wide  ? 

15.  How  many  hours  daily  ought  30  men  to   labor   to 
perform  in  10  days  a  piece  of  work  which  is  f  as  great 
as  a  similar  job  which  25  men,  working  12  hr.  per  day, 
accomplished  in  12  days  ? 

16.  If  $475  yield   $171  interest  in  6  years,  how  long 
will  it  take  $960  to  double  itself  at  the  same  rate  ? 

17.  A  bin  8  ft.  long,  6  ft.  wide,  and  4^  ft.  deep  will  con- 
tain 270  bushels  of  wheat.    How  deep  must  another  bin  be 
built,  that  is  12  ft.  long  and  9  ft.  wide,  to  hold  405  bushels  ? 

18.    How  many  days  ought  it  to  take  9  men  to  build  a 

wall  350  feet  long,  2£  feet  high,  and  3  ft.  thick,  if  10  men 
build  a  wall  312  ft.  long,  6  ft.  high,  and  2  ft.  thick,  in  30 
days  ? 


198  SENIOR    ARITHMETIC. 

PARTNERSHIP. 

336.  Oral. 

1.  Charles  and  John  share  $28  in  the  ratio  of  2  to  5. 
How  much  has  each  ? 

SOLUTION.  —  Charles  has  $2  as  often  as  John  has  $5.  Both  have 
-$7.  Charles  has  f  and  John  S  of  $7.  Since  they  have  respec- 
tively f  and  f  of  a  part  of  $28,  they  must  have  the  same  fractions 
of  the  whole.  Therefore, 

Charles  has  \  of  $28  =    $8.  j    Ang 

John      has  f  of  $28  =  $20.  / 

2.  Divide  30  into  two  such  parts  as  shall  be  to  each 
other  as  7  to  8. 

3.  A  man  and  a  boy  together  earn  $48.     The  man  has 
earned  $3  to  the  boy's  $1.     What  is  each  one's  share  ? 

4.  A   horse   and   a  cow   were  bought  for  $150.     The 
horse  cost  twice  as  much  as  the  cow.     What  was  the  cost 
of  each  ? 

5.  Divide    140  into    four  parts  that  shall   be  to  each 
other  as  2,  3,  4,  and  5. 

6.  A   father  divided  $7200   among  his  three   sons  in 
proportion  to  their  ages,  which  were  10,  12,  and  14  years 
respectively.     What  was  the  share  of  each  ? 

7.  A  man  .divided  $3.60  among  three  boys,  giving  to  the 
first  5  cents  as  often  as  he  gave  6  cents  to  the  second  and 
7  cents  to  the  third.     How  much  did  each  boy  receive  ? 

8.  Professor  Adams  caught  520  fish  in  a  season,  con- 
sisting of  trout,  black  bass,  and  pickerel,  in  the  proportion 
of  5,  4|,  and  3|.     How  many  of  each  kind  did  he  catch  ? 

337.  The  association  of  two  or  more  persons  in  business 
is  called  Partnership. 

The  persons  associated  are  Partners. 


PARTNERSHIP.  199 

The  association  is  called  a  Firm,  or  Company. 
All   the   money   or  property  furnished  by  the  partners 
constitutes  the  Capital. 

338.    To  find  each  partner's  share  of  the  Gain  or  Loss,  when 
their  capital  is  employed  for  the  same  time. 

1.  A,  B,  and  C  formed  a  partnership.  A  contributed 
to  the  capital  $800  ;  B,  $1000  ;  and  C,  $1200.  At  the 
end  of  a  year  they  found  that  there  was  a  gain  of  $1500. 
What  was  each  man's  share  of  the  gain  ? 

SOLUTION.  — 

$800  +  $1000  +  $1200  =  $3000,  entire  capital. 
A's  gain,  3«o7o,  or  ^  of  $1500  -  $400. 
B's  gain,  igg§,  or  T5S  of  $1500  =  $500. 
C's  gain,  £§{$,  or  &  of  $1500  =  $600. 


Rule.  —  Take  for  each  man's  share  of  the  (jain  or  loss 
such  a  part  of  the  whole  gain  or  loss  as  his  capital  is 
of  the  whole  capital. 

2.  Mr.  Wilson  and  Mr.  Mead  entered  into  partnership. 
Mr.  Wilson's  capital  was  $3000,  and  Mr.  Mead's  $2000. 
They  gained  $1500.     What  was   each    partner's  share  of 
the  gain  ? 

3.  Messrs.  Jones  and  Smith  are  partners,  with  a  capital 
of  $3000  and   $5000   respectively.     After  one  year  they 
find  that  they  have  gained  $2000.     How  much  of  the  gain 
should  each  receive  ? 

4.  Three   men    form   a  partnership  at  the  same   time. 
A  invests  $1250;  B,  $2000;  C,  $1550.     They  gain  $1200. 
What  is  each  man's  share  of  the  gain  ? 

5.  Three  men  hired  a  coach  to  convey  them  to  their 
respective  homes.     A's  home  was  20  miles  away,  B's  24 
miles,  and   C's  28   miles.     They  paid  $24  for  the  coach. 
What  ought  each  to  pay  ? 


200  SEN1OU    ARITHMETIC1. 

6.  A  cargo  of  wheat  valued  at  $4500  was  entirely  de- 
stroyed.    One-third  of  it  belonged  to  A,  two-fifths  to  B, 
and  the  remainder  to  C.     What  was  each  one's  share  of 
the  loss,  there  being  an  insurance  of  $3600  ? 

7.  A  man  fails  in  business  to  the  amount  of  $15000, 
and  his  available  means  amount  to  only  $9000.     How  much 
will  two  of  his  creditors  receive,  to  one  of  whom  he  owes 
$3000,  and  the  other  $4500  ? 

8.  A  and  B  gain  in  business  $2500,  of  which  A's  share 
is  $1000,  and  B's  $1500.     What  part  of  the  capital  does 
each  furnish  ?  and  what  is  the  investment  of  each  if  their 
joint  capital  is  $16000  ? 

9.  I  form  a  partnership  with  two  members  of  my  class. 
The  second  member  invests  a  certain  amount,  the  first  in- 
vests ^  as  much,  while  I  invest  as  much  as  the  other  two. 
What  share  of  the  profit  do  I  get  ? 

10.  Purchased  a  flour-mill  for  $42000.     X's  share  of  the 
mill  was  T52,  Y's  J,  and  Z's  the  remainder.     At  the  end  of 
three  years   they  sold  the   mill  at  a  reduction  of  $5000, 
but  the  profits  in  the  business  during  the  three  years  were 
$20000.     What  was  each  man's  net  gain  ? 

11.  Two  persons  have   invested   in   trade   $800.     They 
gain   $150.     The  gain  and   stock   of  the   first   amount  to 
$570.     What  is  the  stock  and  the  gain  of  each  ? 

When  the  capital  of  the  partners  is  not  employed  for  the  same 
time. 

A  and  B  formed  a  partnership.  A  furnished  $500  for 
8  months,  and  B  $600  for  10  months.  They  gained  $360. 
What  was  each  partner's  gain  ? 

SOLUTION.  —  A  $500  x    8  mo.  =  $4000  for  1  mo. 
B  $600  x  10  mo.  =    6000  for  1  mo. 

si  0000 


PAKTNERSHIP.  20l 

A's  share  =  T4o  of  $3(30  =  $144. 
B's  share  =  &  of  $360  =  $210. 

The  use  of  $500  for  8  months  is  equivalent  to  the  use  of  $4000  for  1 
month ;  and  the  use  of  $000  for  10  months  is  equivalent  to  the  use 
of  $6000  for  1  month.  Consider  A's  capital  to  be  $4000  and  B's 
$0000  =  A's  share  of  gain,  j45;  B's  share  of  gain,  T60-. 

2.  A  commenced  business  with  $10000  capital.     Four 
months  later  B  put  in  $10500.     Their  profits  at  the  end  of 
a  year  were  $5100.     What  was  each  man's  share  of  the 
gain  '/ 

3.  Three  persons  loaned  a  sum  of  money  for  which  they 
received  $1596  interest.     The  first  had  $4000  invested  for 
12  mo.,  the  second  $3000  for  15  mo.,  and  the  third  $5000 
for  8  mo.     How  much  interest  did  each  receive  ? 

4.  A  and  B  were  in  partnership  for  2  years.     A  at  first 
invested  $2000,  and  B  $2800.     At  the  end  of  9  months  A 
took  out  $700,  and  B  put  in  $500.     They  lost  in  the  two 
years  $3720.     Apportion  the  loss. 

5.  A,  C,  and  H  form  partnership.     A  puts  in  $8000, 
C  $5000,  H  $10000.     A's  capital  remains  in  the  business 
8  mo.,  C's  9  mo.,    H's  12  mo.     The   net   gain  is   $6900. 
Find  each  man's  share  of  the  gain. 

6.  Two  partners  entered  business,  agreeing  to  continue 
for  18  months.  A  put  in  $2000  at  first,  and  8  months 
later  $1200  additional.  B  at  first  put  in  $3000,  but  at  the 
end  of  4  months  drew  out  $600.  On  closing  their  account 
they  found  they  had  made  $2808.  What  was  each  man's 
share  of  the  gain  ? 

7.  On  Feb.  1  Messrs.  Scott  and  White  commenced  bus- 
iness with  $3000,  each  furnishing  $1500.  On  April  1 
White  put  in  $1300  more.  On  May  1  they  took  Watson 
into  partnership  with  $2500.  At  the  close  of  the  year, 
how  should  a  net  gain  of  $2400  be  apportioned? 


202  SENIOR    ARITHMETIC. 

8.  A's  capital  was  in  business  6  months,  B's  7  months, 
and  C's  11  months.     A's  gain  was  $600,  B's  $1400,  and  C's 
$990.      Their  joint  capital  was  $7800.     What  was  each 
man's  capital? 

9.  A  put  $600  in  trade  for  5  months,  and  B  $700  for  6 
months.     They  gained  $228.    What  was  each  man's  share  ? 

10.  April  1,  1895,  A  goes  into  business  with  a  capital  of 
$6000 ;  July  1,  he  takes  in  B  as  a  partner  with  a  capital 
of  $8000 ;  and  Oct.  1, 1896,  they  have  gained  $2900.  Find 
the  gain  of  each. 

— - 11.  Three  men,  A,  B,  and  C,  hire  a  pasture  for  6  months 
for  $75.  A  puts  in  10  cows  at  first,  but  at  the  end  of  1 
month  takes  away  4.  B  puts  in  8,  and  in  3  months  takes 
out  5,  but  adds  2  after  2  months  more.  C  puts  in  6,  and 
in  4  months  he  puts  in  8  more.  What  should  each  pay  ? 

QUESTIONS. 

339.      1.    Define  ratio  ;  the  terms  of  a  ratio. 

2.  How  is   ratio  found  ?     What  is  direct  ratio  ?     In- 
verse ratio  ?     A  simple  ratio  ?     A  compound  ratio  ? 

3.  Tell  how  to  find  ratio  when  antecedent  and  conse- 
quent are  given.     To  find  consequent  when  antecedent  and 
ratio  are  given.     To  find  antecedent  when  consequent  and 
ratio  are  given.     What  are  the  principles  of  ratio  ? 

4.  Define  proportion.      How  many  terms  in  a  simple 
proportion  ?     Name  them. 

5.  Give  the  principles  of  proportion. 

6.  What  number  is  placed  for  the  third  term  ?     The 
second  ?     The  first  ?    How  is  the  fourth  term  then  found  ? 

7.  What  is  a  compound  proportion  ?     What  number  is 
placed   as    the  third   term  ?     How  is  each   couplet  then 
arranged  ?     How  find  the  fourth  term  ? 


INVOLUTION.  203 

8.  What  is  a  partnership  ?     A  company  ? 

9.  Define  capital  stock ;  dividends. 

10.  Tell  how  to  find  each  partner's  share  when  the 
capital  of  each  is  invested  for  the  same  time.  When  the 
capital  of  each  is  not  invested  for  the  same  time. 

INVOLUTION. 

340.  1.   3x4x2  =  what  ? 

2.  3x3x3=  what  ? 

NOTE.  —  In  Ex.  2  the  factors  are  equal ;  in  Ex.  1  they  are  unequal. 

The  product  of  equal  factors  is  a  Power. 

3.  What  is  the  product  of  4  taken  3  times  as  a  factor  ? 

4.  What  is  the  product  of  6  taken  twice  as  a  factor  ? 

5.  What  is   the  product  of   §   used   three  times  as  a 
factor  ? 

6.  What  is  the  product  of  .6  used  twice  as  a  factor  ? 

341.  The  process  of  finding  powers  is  Involution. 

342.  A  power  is  named  according  to  the  number  of  its 
equal  factors. 

The  product  of  two  equal  factors  is  the  Second  Power,  or 
Square,  of  the  equal  factor. 

The  product  of  three  equal  factors  is  the  Third  Power,  or 
Cube,  of  the  factor. 

NOTE.  — The  second  power  is  called  a  square  because  the  area  of 
any  square  figure  is  the  product  of  two  equal  factors,  length  and 
breadth;  and  the  third  power  is  called  a  cube  because  the  solidity  of 
any  cube  is  the  product  of  three  equal  factors,  length,  breadth,  and 
thickness. 

343.  A  small  figure  at  the  right  and  above  a  number  to 
show  how  many  times  it  is  to  be  used  as  a  factor  is  called 
an  Exponent.     Thus, 


204  SENIOR    ARITHMKT1C. 

42  =  4  x  4  is  4  to  the  second  power,  or  the  square  of  4  ; 
23  =  2  x  2  x  2  is  2  to  the  third  power,  or  the  cube  of  '2  ; 
3*  =  3  x  3  x  3  x  3  is  3  to  the  fourth  power,  or  the  fourth  power 
of  3. 

Bead:  82,  15»,  57,   (3/4)«,  3/42,  3*/4,  8410,  163. 
344.    Find  the  powers  : 


7.    53. 

10.    65. 

13.    2.58. 

16.    (|)3. 

8.    24. 

11.    I4. 

14.    1.1«. 

17.    (f)«. 

9.    25-. 

12.    .Ol4. 

15.    .0023. 

is.  my 

EVOLUTION. 

345.  l.    What  factor  is  used  3  times  to  produce  27? 

2.  What  are  the  two  equal  factors  of  64  ? 

3.  What  is  one  of  the  three  equal  factors  of  8? 

4.  36  is  the  square  of  what  number  ? 

5.  64  is  the  cube  of  what  number  ? 

6.  144  is  the  second  power  of  what  ? 

7.  1728  is  the  cube  of  what  ? 

346.  One  of  the  equal  factors  of  a  power  is  a  Root. 

One  of  two  equal  factors  of  a  number  is  the  Square  Root 
of  it- 

One  of  the  three  equal  factors  of  a  number  is  the  Cube 
Root  of  it. 

The  fourth  root  of  a  number  is  one  of  its  4  equal  factors. 

The  square  root  of  16  =  4.  The  cube  root  of  27  =  3. 
The  fourth  root  of  16  =  2. 

347.  The  Radical  Sign  (V  )  placed  before  a  number  indi- 
cates that  its  root  is  to  be  found. 

The  radical  sign  alone  before  a  number  indicates  the  square 
root  ;  thus,  V9  =  3  is  read,  the  square  root  of  9  =  3. 


SQUARE    ROOT.  (J2Q5 

348.  A  small  figure  placed  in  the  opening  of  the  radical 
sign  is  called  the  Index  of  the  root,  and  shows  what  root  is 
to  be  taken  ;  thus,  -\/8  =  2  is  read,  the  cube  root  of  8  is  2. 

Read  the  following : 

V81,   A/64,   A/81,  V144,   A/1728,   A/9,   A/<^64. 

EVOLUTION   AND    INVOLUTION. 

349.  1.    Find  the  square  of  11.     The  cube  of  6.     The 
fourth  power  of  5. 

2.  Find   the  square   root  of  49.     The  cube  root  of  8. 
The  square  root  of  T9g. 

3.  92  =  ?     A/9  =  ?     83  =  ?     ^/8  =  ? 

4.  Write  all  the  squares  from  1  to  100. 

5.  Write  all  the  cubes  from  1  to  1000. 

6.  Learn  the  second  and  third  powers  of  numbers  from 
1  to  12. 

SQUARE   ROOT. 

350.  The   square  of  a   number  is   the  product  of  that 
number  taken  twice  as  a  factor. 

Blackboard. 

12=1.  10- =  100.  1002  =  10000. 

92  =  81.  90*  =  8100.  9002  =  810000. 

From  the  above  illustration  it  is  seen  that  annexing  one 
cipher  to  a  number  annexes  two  ciphers  to  the  square  of 
that  number,  as  in  I'2  =  1  ;  10'2  =  100;  100'2  =  10000. 

351.  A   square    contains    twice    as  many  figures  as  its 
root,  or  twice  as  many  less  one. 

Squares  of  even  tens. 
Oral. 

1.  202  =  ?   3.  80'2  =  ?  5.  702  =  ?  7.  500'2  =  ?  9.  6002  =  ? 

2.  502  =  ?  4.  30'2  =  ?    6.  2002  =  ?  8.    9002  =  ? 


206  SENlOli    ARITHMETIC. 

352.  The  square  of   a  number  composed  of   tens  and 
units  may  be  found  as  follows : 

24  =  20  +  4  =  2  tens  +  4  units. 
242  =(20 +  4)  x  (20  +  4). 

20  +  4  =    24 

20  +  4  =    24 

(20  x  4)  +  4  =    96 

202  +  (20  x  4)  =  480 

202  +  2  x  (20  x  4)  +  42  =  576 

From  the  operation,  we  find  that, 

The  square  of  the  tens     .     .  20  =  400 

2  times  the  tens  by  the  units,     2x(20x4)  =  160 
The  square  of  the  units    .     .  4  =    16 

400  +  160  +  16  =  576 

353.  PRINCIPLE.  —  The  square  of  a  number  composed  of 
tens  and  units  is  equal  to  the  square  of  the  tens,  plus  twice 
the  product  of  the  tens  by  the  units,  plus  the  square  of  the 
units. 

FORMULA.  — Tens2  +  2  x  tens  x  units  +  units2. 

Separate  the  following  into  tens  and  units,  and-  find  their 
squares  :  15,  25,  74. 

354.  By  reversing  the  process  we  may  find  the  Square 
Root. 

10.    What  is  the  square  root  of  1225  ? 

SOLUTION.  —  Separating  into  periods  of  two  figures  each,  begin- 
ning at  units,  we  have  12'25.  Since  there  are  two  periods  in  the 
power,  there  must  be  two  figures  in  the  root,  tens  and  units. 

The  greatest  square  of  even  tens  contained  in  1225  is  900,  and 

its  square  root  is  30  (3  tens). 

1225  |  30  4-  5  =  35. 

Tens2,  302  =         900 

2  x  tens  =  2  x  30  =60    325 

2  x  tens  +  units  =  2  x  30  +  5  =  65  |325 

Subtracting  the  square  of  the  tens,  900,  the  remainder  consists  of 
2  X  (tens  x  units)  +  units. 


SQUARE    KOOT. 


207 


325,  therefore,  is  composed  of  two  factors,  units  being  one  of 
them,  and  2  x  tens  +  units  being  the  other.  But  the  greater  part 
of  this  factor  is  2  x  tens  (2  x  30  =  60).  By  trial  we  divide  325  by 
60  to  find  the  other  factor  (units),  which  is  5,  if  correct.  Complet- 
ing the  factor,  we  have  2  x  tens  +  units  =  65,  which,  multiplied  by 
the  other  factor,  5,  gives  325,  proving  the  correctness  of  the  solution. 
Therefore  the  square  root  is  30  +  .">  =  35. 

355.  Square  root  may  be  explained  by  the  aid  of  dia- 
grams. 

The  area  of  every  square  surface  is  the  product  of  two 
equal  factors,  length  and  width. 

Finding  the  square  root  of  a  number,  therefore,  is  equiv- 
alent to  finding  the  width  of  a  square  surface,  its  area 
being  given. 

356.  The    following    formulas    illustrate  the  principles 
which  underlie  the  operations  of  square  root : 

1.  Length  x  Width  =  Area. 

2.  Area  4-  Length  =  Width. 

3.  Area  -i-  Width  =  Length. 

1.    Find  the  width    of   a    square  whose    area   is  1296 
sq-  ft.  Fig.  7. 


AREA. 
1296 

900 
396 
396 


30ft. 
6ft. 


30  ft. 


30  ft. 


6ft. 


2  x  30  ft.  =  60  ft. 

2  x  30  ft.  +  6  ft.  =  66  ft. 

SOLUTION. 
The  greatest  square  of 

even    tens    contained    in      ^_ ^_^ 

1296  sq.  ft.  is  900  sq.  ft.      |  b  |  d  \c\ 

(Squared).    Its  width  is  Fig.  2. 

30  ft.     1296  sq.  ft.  -  900 

sq.  ft.  =  396  sq.  ft.,  the  area  of  6,  c,  and  d,  considered  as  one  rec- 
tangle (Fig.  2),  whose  width  we  desire  to  find.  The  length  of  this 
rectangle  is  (2  x  30  ft.)  60  ft.  +  the  length  of  c.  But  we  cannot 


208  SENIOR    ARITHMETIC. 

know  the  length  of  c  till  we  find  its  width.  By  trial  (Formula  2), 
we  divide  the  area,  396  sq.  ft.,  by  60  ft.,  its  approximate  length. 
The  quotient,  if  correct,  is  6  ft.,  the  width  desired.  To  test  the  cor- 
rectness :  Add  the  6  ft.  to  the  trial  divisor,  and  we  have  66  ft.,  the 
entire  length  of  a,  />,  and  c,  which  (Formula  1),  multiplied  by  its 
width,  6  ft.,  gives  its  area,  396  sq.  ft.  There  is  no  remainder,  and 
the  work  is  correct.  Therefore, 

30  ft.,  the  width  of  A,  +  6  ft.,  the  width  of  a,  7>,  and  c,  =  36  ft., 
the  width  of  the  original  square. 

NOTES.  —  All  the  numbers  in  the  middle  column  denote  area. 
1296  sq.  ft.  —  area  of  the  original  square;  900  sq.  ft.  =  the  area  of 
A ;  and  396  sq.  ft.  the  area  of  b,  c,  and  d. 

•  The  numbers  in  the  left-hand  column  denote  length.  60  ft.  —  the 
approximate  length  (or  the  trial  divisor)  of  b,  c,  and  d  ;  and  66  ft. 
the  exact  length,  or  the  complete  divisor. 

The  numbers  in  the  right-hand  column  denote  width.  30ft  =  the 
width  of  A  •  and  6  ft.  the  width  of  b,  c,  and  d  ;  36  ft.  =  the  width  of 
the  original  square. 

In  dividing,  to  find  the  width  of  b,  c,  and  d,  since  the  divisor  is 
too  small,  care  must  be  taken  that  the  quotient  figure  be  not  too 
large. 

SHORT  METHOD. 

357.    Ex.  2.    Find  the  square  root  of  1,306.0996. 

i3W.09'96  (36.14 

9 

66j   406 
396 

721 )  1009 
721 


7224)  28896 
28896 

Rule.  —  Beginning  at  the  decimal  point,  separate  the  num- 
ber into  periods  of  two  figures  each,  pointing  whole 
numbers  to  the  left  and  decimals  to  the  right.  Find 
the  greatest  square  hi  the  left-hand  period,  and  irrlte  its 


SQUATJR   ROOTV  209 

root  at  the  rigid.  Subtract  the  square  from  the  left-hand 
period,  and  briny  down  the  next  period  for  a  dividend. 

Divide  the  dividend  by  twice  the  root  already  found,  and 
annex  the  quotient  to  the  root,  and  to  the  divisor. 
Multiply  this  complete  divisor  by  the  last  root  figure, 
and  bring  down  the  next  period  for  a  dividend,  as 
before. 

Proceed  in  this  manner  till  all  the  periods  are  exhausted. 

NOTE  1.  —  When  0  occurs  in  the  root,  annex  0  to  the  trial  divi- 
sor, bring  down  the  next  period,  and  divide  as  before. 

NOTE  2. —  If  there  is  a  remainder  after  all  the  periods  are  ex- 
hausted, annex  decimal  periods. 

NOTE  3.  —  If,  after  multiplying  by  any  root  figure,  the  product  is 
larger  than  the  dividend,  the  root  figure  is  too  large  and  must  be 
diminished.  Also  the  last  figure  in  the  complete  divisor  must  be 
diminished. 

NOTE  4.  —  For  every  decimal  period  in  the  power,  there  must  be 
a  decimal  figure  in  the  root. 

NOTE  5.  —  If  the  last  decimal  period  does  not  contain  three  fig- 
ures, supply  the  deficiency  by  annexing  one  or  more  ciphers. 

Ex.  3.    Find  the  square  root  of  253009. 

SOLUTION. 

25'30'09(j>          As   0   occurs    in    the  25'30'09  (_503  An*. 

25  root,  we  annex  0  to  the  25 

10)      30  trial  divisor,  10,  and  an-     1003)     3000 

other  period  to  the  divi-  3009 

dend,  and  divide  as  before.     Thus,  — 

NOTE.  —  To  find  the  square  root  of  a  common  fraction,  extract 
the  root  of  each  term  separately.  If  both  terms  are  not  squares, 
change  the  fraction  to  a  decimal,  and  then  extract  the  root.  The 
result  will  be  the  approximate  root.  Change  mixed  numbers  to 
improper  fractions. 

4.    What  is  the  square  root  of  T8J?  ?  =  ^  • 

V144 


210  SENIOR   ARITHMETIC. 

Find  the  square  root  of : 

5.  8836.  14.    .06432  23. 

6.  15876.  15.    .005625  24.  f-£ 

7.  370881  16.    .913936  25.  fj$f 

8.  46656  17.    25.6036  26.  4i 

9.  820836  18.    24.3049  27.  ^ 

10.  29.0521  19.  .612089  28.  §ff£ 

11.  9.2416  20.  329.7643217  29.  36.45? 

12.  3180.96  21.  1684.298431  30.  2863y 

13.  .007921  22.  389765268  31.  189  Jfj 

Find  the  square  root  to  four  decimal  places : 

32.  .15  36.    72.5  40.    963 

33.  .18  37.    119  41.    13.2f 

34.  17  38.    3.67  42.    .009T\ 

35.  4.7  39.    .222  43.    .003f 

44.-  What  is  the  length  of  one  side  of  a  square  field 
that  has  an  area  equal  to  a  field  75  rd.  long  and  45  rd. 
wide  ? 

45.    How  wide  is  a  field  containing  7056  square  rods  ? 

Perform  the  indicated  operations. 

NOTE.  —  Carry  decimals  to  the  third  place. 


46.  V3.26  X  .0063.  51.  V3.532  -4-  6.28. 

47.  .OSxVfTf  52.  vT+~6*T2. 

1     ... 

53.  v  5-    +  (I )  . 

54.  V625  +  1296. 


55.    V625~+  V1296. 


RIGHT-ANGLED   TKI  ANGLES. 


211 


RIGHT-ANGLED    T1UA XGLES. 

358.  A  triangle  having  one  right  angle  is  a  Right-Angled 
Triangle. 

359.  The  side  opposite  the  right  angle  is  the  Hypothe- 
nuse,  as  AB.     BC  is  the  Perpendicular,  and  AC  the  Base. 
In   the    triangle    ABC,    the 

hypothenuse  is  5  inches,  the 
perpendicular  3  inches,  and 
the  base  4  inches. 


360.  It  will  be  seen  that  the  square  of  the  hypothenuse 
is  25  sq.  in.,  which  is  equal  to  the  square  of  the  base,  16 
sq.  in.,  plus  the  square  of  the  perpendicular,  9  sq.  in. 

PRINCIPLE.  —  The  square  of  the  hypothenuse  equals  the 
sura  of  the  squares  of  the  two  shorter  sides.  Therefore, 
to  find  the  hypothenuse,  take  the  square  root  of  the  sum 
of  the  squares  of  the  base  and  perpendicular. 


-\/  Base'-2  -f  Perpendicular2  =  Hypothenuse. 

361.  To  find  the  base  or  the  perpendicular,  take  the  square 
root  of  the  difference  between  the  squares  of  the  hypothe- 
nuse and  the  other  side. 


Hypothenuse2  —  Base2  =  Perpendicular. 


•\/  Hypothenuse2  —  Perpendicular-1  =  Base. 


SENIOR    ARITHMETIC. 


1.    The  base  of  a  right-angled  triangle  is  32  ft,  and  the 
perpendicular  24  ft.     What  is  the  hypothenuse  ? 


SOLUTION.  —  322  +  242  =  1000.         A/1600  =  40  ft.     Ans. 


Or,  v/822  +  24*  -  40  ft. 

2.    The  hypothenuse  of  a  right-angled  triangle  is  40  ft., 
and  the  base  32  ft.     What  is  the  perpendicular  ? 


402  _  322  =  576  V576  =  24  ft.     Ana. 


Or,  A/40*  -  32*  =  24  ft. 

3.  A  40-foot  ladder  placed  24  feet  from  a  house  will 
just  reach  to  the  top  of  it.     How  high  is  the  house  ? 

4.  What  is  the  length  of  a  ladder  that  will  reach  the 
top  of  a  house  40  feet  high,  when  the  foot  is  placed  30  feet 
from  the  house  ? 

5.  A  rope  150  ft.  long  fastened  to  the  top  of  a  flag-pole 
reaches  the  ground  40  feet  from  the  base.      How  high  is 
the  pole  ? 

6.  What  is  the  hypothenuse  of  a  right-angled  triangle 
whose  perpendicular  is  36  feet,  and  whose  base  is  27  feet? 

7.  A  square  farm  contains  360  acres.     What  is  the  di- 
agonal distance  between  its  opposite  corners  ? 

8.  A  telegraph  pole  32  feet  high  casts  a  shadow  28  feet 
in  length.     What  is  the  distance  from  the  top  of  the  pole 
to  the  end  of  the  shadow  ? 

9.  The  base  of  a  right-angled  triangle  is  16  in.,  and  the 
perpendicular  is  12.8  m.     What  is  the  hypothenuse  ? 

10.  A  boy  rides  his  wheel  due  north  at  the  rate  of  15 
miles  an  hour,  and  another  boy  starting  from  the  same 
place,  rides  (hie  east  at  the  rate  of  18  miles  an  hour.  How 
far  are  they  apart  at  the  end  of  5  hours  ? 


SIMILAR    SURFACES.  213 

11.  What  is  the  length  of  the  diagonal  of  a  room  10  ft. 
long  and  12  ft.  wide  ? 

12.  A  crayon  box  is  6  in.  long,  4  in.  wide,  and  4  in.  high. 
What  is  the  diagonal  distance  across  the.  bottom,  between 
the  opposite  corners  ? 

13.  A  street  is  32  ft.  wide  from  curb  to  curb.     A  tele- 
graph pole  40  ft.  high  stands  upon  one  side  of  the  street. 
How  long  must  a  wire  be  to  reach  from  the  top  of  the  pole 
to  the  opposite  side  of  the  street  at  the  curb  ? 

SI  MIL  A  R    S  URFA  CES. 

362.  Surfaces  having  the  same  form  without  regard  to 
size  are  Similar  Surfaces. 

NOTE.  — Any  two  squares  or  any  two  circles  of  different  size  are 
Similar  Figures.  Rectangles,  triangles,  etc.,  are  similar  when  their 
corresponding  dimensions  are  proportional. 

Oral. 

1.  What  is  the  area  of  a  square  whose  side  is  2  ft.  ? 

2.  AVhat  is  the  area  of  a  square  whose  side  is  3  ft.  ? 

3.  AVhat  is  the  ratio  of  the  two  sides  ? 

4.  What  is  the  ratio  of  the  two  areas  ? 

5.  Are  these  ratios  equal  ?     (2  ft, :  3  ft*)     (4  sq.  ft. :  9 
sq.  ft.) 

SOLUTION.  —  From  the  illustration  it  will 
be  seen  that  the  areas  are  to  each  other  as 
the  squares  of  the  sides ;  not  as  2  to  3,  but 
as  4  to  9. 

PRINCIPLES.  —  Similar  surfaces  are  to  each  other  as  the 
squares  of  their  corresponding  dimensions. 

Corresponding  dimensions  are  to  each  other  as  the  square 
roots  of  their  areas. 


214  SENIOR   ARITHMETIC. 

6.  A  circle  is  4  inches  in  diameter ;  another  is  8  inches 
in  diameter.     What  is  the  ratio  of  their  areas  ? 

7.  A  circle  has  an  area  of  16  square  feet ;  another  has 
an  area  of  64   sguare  feet.     What  is  the   ratio  of  their 
diameters  ? 

8.  The  area  of  a  rectangle  12  ft.  long  is  84  square  feet. 
What  is  the  area  of  a  similar  rectangle  6  feet  long  ? 

9.  Two  similar  fields  have  areas  of  12  acres  and  8  acres 
respectively ;   the  larger  is  32  rods  wide  ?     How  wide  is 
the  smaller  ? 

10.  The  altitudes  of  two  similar  triangles  are  20  ft.  and 
10  ft. ;  the  area  of  the  smaller  is  80  square  feet.  What  is 
the  area  of  the  larger  ? 

CUBE   ROOT. 

363.  The  cube  of  a  number  is  the  product  of  that  num- 
ber taken  three  times  as  a  factor. 

Blackboard. 

13  =  l.  103  =  1000.  10Q3  =  1000000. 

93  =  729.         903  =  729000.         9003  =  729000000. 

364.  Annexing  one  cipher  to  a  number,  annexes  three 
ciphers  to  the  cube  of  the  number,  as  shown   in  I3,  103, 
100s,  etc. 

365.  Cubes  of  even  tens. 

1.  103  =  ?  4.    403  =  ?  7.    3003  =  ? 

2.  303  =  ?  5.    80s  =  ?  8.    8003  =  ? 

3.  503  =  ?  6.    2003  =  ?  9.    9003  =  ? 

366.  The  cube  of  a  number  composed  of  tens  and  units 
may  be  found  as  follows : 


CUBE    BOOT.  215 

24   =  20  -f-  4  =  2  tens  +  4  units ; 

24*  =  (20  +  4)  x  (20  +  4)  x  (20  +  4). 


20  +  4  = 
20  +  4  = 

24 
24 

96 

480 

(20  X  4)  +  42  - 
202  +  (20  x  4)     = 

202  +  2  x  (20  x  4)  +  42  = 
20  +  4  = 

576 
24 

(202  x  4)  +  2  x  (20  x  42)  +  43  =    2304 
20^  +  2  (202  X  4)  +  (20  x  42)  =    1152 

203  +  3  x  (202  x  4)  +  3  x  (20  x  42)  +  43  -  13824 

From  the  operation  we  find  that, 

The  cube  of  the  tens 203=  8000 

3  times  the  square  of  tens  by  units     .     .     .  3  (202  x  4)  =  4800 

3  times  the  tens  by  the  square  of  the  units,  3  (20  x  42)  =  960 

The  cube  of  the  units 43  =  64 

8000  +  4800  +  960  +  64  =  13824 

367.  PRINCIPLE.  —  The  cube  of  a  number  composed  of 
tens  and  units  is  equal  to  the  cube  of  the  tens  plus  3  times 
the  square  of  the  tens  by  the  units,  plus  3  times  the  tens 
by  the  square  of  the  units,  plus  the  cube  of  the  units. 

FORMULA.  —  Tens3  +  3  X  tens2  X  units  -f  3  X  tens  X 
units'2  -f-  units3. 

10.  Separate  the  following  into  tens  and  units,  and  find 
their  cubes  :  35,  54,  63. 

368.  By  reversing  the  process,  we   may  find   the   cube 
root. 

11.  What  is  the  cube  root  of  13824  ? 

SOLUTION.  —  Separating  into  periods  of  three  figures  each,  begin- 
ning at  units,  we  have  13'824.  Since  there  are  two  periods  in  the 
power,  there  must  be  two  figures  in  the  root,  tens  and  units. 


210  SENIOR    ARITHMETIC. 

The  greatest  cube  of  even  tens  contained  in  13824  is  8000,  and  its 
cube  root  is  20  (2  tens). 

13'824  I  20  +  4 

Tens3  =  205  =  8000  ~ 

:J  x  tens-  =  3  x  20-  =  1200        5824 
3  X  tens  X  units  =  3  X  (20  x  4)  =    240 
Units'2  =  4-  =      10 

3  x  tens'2  +  3  tens  x  units  +  units'2  =  1456 
(3  X  tens'2  +  3  x  tens  x  units  +  units'2)  x  units  =  5824 

Subtracting  the  cube  of  the  tens,  8000,  the  remainder,  5824,  con- 
sists of  3  x  (tens'2  x  units)  +  3  x  (tens  x  units'2)  +  units3.  5824. 
therefore,  is  composed  of  two  factors,  units  being  one  of  them,  and 
3  x  tens'2  +  3  x  tens  x  units  +  units2,  being  the  other.  But  the 
greater  part  of  this  factor  is  3  x  tens'2.  By  trial  \ve  divide  5824 
by  3  x  tens2  (1200)  to  find  the  other  factor  (units),  which  is  4  if 
correct.  Completing  the  divisor,  we  have  1200-  +  3  x  (20  -f  4)  +  4- 
=  1456,  which,  multiplied  by  the  units,  4,  gives  the  product,  5824 
proving  the  correctness  of  the  work.  Therefore  the  cube  root  is 
20  +  4  -  24. 

369.    To  find  the  cube  root  by  the  aid  of  blocks. 

Finding  the  cube  root  of  a  number  is  equivalent  to  find- 
ing the  thickness  of  a  cube,  its  volume  being  given. 

The  following  formulas  illustrate  the  principles  that 
underlie  operations  in  cube  root. 

NOTE.  _  For  convenience  Z,  6,  t,  and  r  will  represent  length, 
breadth,  thickness,  and  volume,  respectively. 

(1)  I  X  b  X  t  =  v.      (2)  V-*-(lX  &)  -  t.     (3)  v  +  (lXt) 

=  b.      (4)  v  ~  (b  X  t)  =  L 

12.  What  is  the  thickness  of  a  cube  whose  volume  is 
13824  cubic  feet  ? 


VOLCMES.   THICKNESS. 

3  x  202         =1200       13'824        20ft.         SOLUTION.  —  The  greatest 
3  x  20  x  4  =    240         8000          4ft.     cube  of  even  tens  contained 
4-2  =      16          5824         24  ft.     in  13824  cu.  ft.  is  8000  cu.  ft. 
1456         5824  (Cube    .4.)      Its   thickness, 

therefore,  is  20  ft.     Subtracting  8000  (.4)  from  13824  leaves  a  re- 
mainder of  5824  cu.  ft.,  which  are  added  in  solids  of  equal  thickir  - 


ess 


CUBU    1400T. 


217 


to  three  sides  of  ^4,  as  seen  in  Fig.  3.  It  now  remains  to  find  the 
thickness  of  the  additions  (6,  c,  d),  O,/,  r/),  and  h,  which  have  a 
uniform  thickness.  As  the  solids  b,  c,  d,  form  the  greater  part  of  the 
volume  of  the  additions  (5824  cu.  ft.),  and  the  length  and  breadth 
of  each  is  20  ft.  (the  length  and  breadth  of  A},  by  trial,  using  For- 
mula 2,  we  find  5824  -f  (3  x  20'-2)  =  4  ft.,  thickness  of  the  additions, 
if  correct.  Knowing  the  thickness,  which  is  also  the  breadth  of  e, /, 
r/,  /<,  we  find  the  product  of  the  length  and  breadth  of  e,  /,  g  =  3  x 


g 


20  x  4  =  240  sq.  ft. ;  and  that  of  h  —  42  =  16  sq.  ft. ;  both  of  which 
added  to  1200  sq.  ft.  =  the  product  of  the  length  and  breadth  of  all 
the  additions.  This  product,  by  Formula  1,  multiplied  by  the  thick- 
ness, 4  ft.,=  5824  cu.  ft.;  proving  the  correctness.  Therefore, 

The  thickness  of  a  cube  whose  volume  is  13824  cu.  ft.  is  20  +  4  ft. 
=  24  ft. 

The  numbers  in  the  middle  column  (Ex.  12;  all  indicate  volume: 
13824  =  volume  of  original  cube. 
8000  =  volume  of  Cube  A. 
5824  =  volume  of  the  additions  (6,  c,  d),  (e,/,  gr),  and  //. 


218  SENIOK    ARITHMETIC . 

The  numbers  in  the  left-hand  column  indicate  product  of  length 

and  breadth  : 

1200  =  I  x  b  of  solids  6,  c,  d. 

240  =  I  x  b  of  solids  e,/,  g. 
16  =  I  x  b  of  cube  /«. 

The  numbers  in  the  right-hand  column  indicate  thickness: 

20  ft.  =  thickness  of  A. 
4  ft.  =  thickness  of  all  the  additions. 
24  ft.  =  thickness  of  original  cube. 

370.    Short  method. 
Rule  for  finding  the  cube  root: 

Beginning  at  the  decimal  point,  separate  the  number  into 
periods  of  three  figures  each  ;  thus:  16'581'.375. 

Find  the  greatest  cube  in  the  left-hand  period,  and  write 
its  root  at  the  right.  Subtract  the  cube  from  the  left- 
hand  period,  and  bring  down  the  next  period  for  a  div- 
idend ;  thus, 


8581 

To  find  the  trial  divisor,  square  the  root  already  found 
with  a  cipher  annexed,  and  multiply  by  3  ;  thus, 


8  20 

Trial  divisor,  1200/8581  _20 

400 

3 

1200 

To  find  the  trial  figure,  find  how  many  times  the  trial 
divisor  is  contained  in  the  dividend  j  thus, 

16'581'.375|_25 
8  20 

Trial  divisor,  1200/8581  _20 

400 

8 

1200 


CUBE    KOOT. 


219 


To  find  the  correction,  multiply  the  former  root  by  3,  an- 
nex the  trial  figure,  and  multiply  by  the  trial  figure  ; 

thus, 

16/581/.375|25.5 
8  2 

3_ 
65 


1200 
325 


Complete  divisor,  1525 
187500 
3775 
191275 


8581 


7625 


956375 


956375 


2 

325 


Continue  thus,  until 
all  the  periods  are-ex- 
hausted. 


NOTE  1. — When  there  is  a  remainder  after  all  the  periods  are 
exhausted,  annex  decimal  periods,  and  continue  the  process  as  far 
as  desired.  The  result  will  he  the  approximate  root. 

NOTE  2.  —  When  a  cipher  occurs  in  the  root,  we  annex  two 
ciphers  to  the  trial  divisor,  and  bring  down  the  next  period. 

NOTE  3.  —  The  right-hand  decimal  period  must  have  three 
places. 

13.    What  is  the  cube  root  of  8.414975304  ? 

OPERATION. 
8.414/975/304  I  2.034 


120000 
1809 

414975 
365427 

121809 

-6362700 
24376 

49548304 
49548304 

12387076 

Since  0  occurs  in  the  root,  an- 
nex 00  to  the  trial  divisor,  mak- 
ing 120000 ;  bring  down  the  next 
period. 


NOTE.  —  To  find  the  cube  root  of  a  common  fraction,  extract  the 
root  of  each  term  separately.  If  both  terms  are  not  cubes,  reduce  to 
a  decimal  and  then  extract  the  root.  The  result  will  be  the  approx- 
imate root. 


Find  the  cube  root  of 

14.  42875. 

15.  884736. 


16.  4492125. 

17.  77854483, 


220  SENIOR    ARITHMETIC. 

18.  8.615125.  21.    1879.080904. 

19.  17.373979.  22.    32.890033664. 

20.  450827.  23.    10077696. 

24.  What  is  the  cube  root  of  JffSSi?  A?  itH?  39^? 

A? 

Extract  the  cube  root  to  the  third  decimal  place : 

25.  14.323.  27.    .06324.  29.    3. 

26.  31982.4.  28.    .0015.  30.    7. 

31.  What  is  the  width  of  a  cube  whose  solidity  is  91125 
cubic  inches  ? 

32.  A  cubical  cistern  holds  50  barrels  of  water.     How 
deep  is  it  ? 

33.  What  is  the  entire  surface  of  a  cube  whose  side  is 
9  ft.? 

34.  -\/S()6  X  32.5  =  ? 

SIMILAR    SOLIDS. 

371.  Solids  having  the  same  form  without  regard  to  size 
are  Similar  Solids.  Any  two  cubes  or  any  two  spheres  are 
similar  solids.  Solids  are  similar  when  their  correspond- 
ing dimensions  are  proportional. 

PRINCIPLES.  —  Similar  solids  are  to  each  other  as  the 
cubes  of  their  corresponding  dimensions. 

The  corresponding  dimensions  of  similar  solids  are  to 
each  other  as  the  cube  roots  of  their  volumes. 

1.  A  globe  is  3  inches  in  diameter,  and  another  6  inches 
in  diameter.     What  is  the  ratio  of  their  volumes  ? 

EXPLANATION.  —  They  are  to  each  other  as  33  to  63  =  27  :  216. 

2.  There  are  64  cubic  inches  in  a  4-inch  cube.     How 
many  in  an  8-inch  cube  ? 


SIMILAR   SOLIDS.  221 

3.  Two  similar  solids  contain  386  and  284  cubic  inches 
respectively.     If  the  larger  is  11  inches  thick,  how  thick 
is  the  smaller  ? 

4.  If  a  man  6  ft.  2  in.  tall  weighs  215  pounds,  what 
should  be  the  weight  of  a  man  5  ft.  10  in.  tall  of  the  same 
proportions  ? 

5.  The  width  of  a  bin  is  4  ft.  6  in.     How  wide  must  a 
similar  bin  be  to  hold  4  times  as  much  ? 

6.  If  an  orange  2^  inches   in  diameter  costs  5  cents, 
what  should  an  orange  3^  inches  in  diameter  cost  ? 

QUESTIONS. 

372.  1.  What  is  involution  ?  A  power  of  a  number  ? 
The  first  power  ?  The  second  power  ?  The  third  power  ? 
What  are  the  second  and  third  powers  called  ?  What  is 
the  exponent  of  a  power  ? 

2.  What  is  evolution  ?  A  root  ?  The  square  root  of  a 
number  ?  The  cube  root  of  a  number  ?  The  fourth  root 
of  a  number  ?  How  is  a  root  indicated  ?  The  square 
root  ?  The  fourth  root  ? 

e^.  3.    Tell  how  to  find  the  side  of  a  square  when  the  area 
is  given  ? 

4.  Tell  how  to  find  the  edge  of  a  cube  when  its  volume 
is  given  ? 

5.  What  kind  of  measure  is  a  cube  ? 

6.  A  cube  contains  how  many  times  as  many  figures  as 
its  root  ? 

What  is  shown  when  the  number  is  separated  into 
periods  of  three  figures  each  ? 

7.  What  is  the  cube  root  of  a  number  ?     Two  answers. 

8.  Cube  the  numbers  from  1  to  10. 


222  SENIOR   ARITHMETIC. 

9.    What  is  the  first  root  figure  ?     What  kind  of  meas- 
ure is  it  ? 

10.  How  is  the  trial  divisor  found  ?     What  is  the  trial 
divisor? 

11.  What  kind  of  measure  is  it?     Why  is  it  a  trial 
divisor  ? 

12.  How  is  the  correction  found  ? 

13.  What  kind  of  measure  is  the  correction  ? 

14.  What  is  the  complete  divisor  ?    What  kind  of  meas- 
ure is  it  ? 

15.  What  is  a  right-angled  triangle  ? 

16.  What   principles    are    true    of    all    right-angled    tri- 
angles ? 

17.  Tell  how  to  find  hypothenuse,  base,  perpendicular. 

18.  What   are    similar   figures  ?      What   principles    are 
true  of  them  ? 

19.  What  are  similar  solids  ?     What  principles  are  true 
of  them  ? 

REVIEW. 
373.    Oral. 

1.    What  is  the  cost  of  20  pounds  of  sugar  at  6§  cents 
a  pound  ? 

_  ~    2.    A  man   owning  f  of  a  farm  sold  ^  of   his  share. 
What  part  does  he  still  own  ? 

3.  A  can  do  a  piece  of  work  in  2  hours,  and  B  in  3 
hours.     In  what  time  can  both  do  it,  working  together  ? 

4.  Two  men  receive  $60  for  painting  a  house.     One 
worked  for  $2  a  day,  and  the  other  $3  a  day.     How  much 
money  should  each  receive  ? 

5.  What  is  the  interest  of  $ 500  for  2£  years  at  6<    ? 


REVIEW.  228 

6.  What  is  the  cost  of  64  straw  hats  at  $1  each  ?     At 

$ .50  ?     $.25  ?     At  $.12i  ?     At  $1.25  ?     At  $2.50  ? 

7.  If  4  oranges  cost  12  cents,  what  will  7  oranges  cost  ? 

8.  If  |  of  a  yd.  of  silk  costs  $1J,  what  will  1J  yards 
cost? 

9.  If  a  man  6  feet  tall  casts  a  shadow  8  feet  long,  how 
long  a  shadow  will  a  boy  4±  feet  tall  cast  ? 

10.  If  f  of  my  money  is  silver  and  the  rest  bills,  and  I 
have  $180,  how  much  of  each  kind  have  I  ? 

11.  If  |  of  a  cord  of  wood  costs  $1.50,  what  will  a  cord 
cost  ?  5  cords  ? 

12.  A  boy  buys  papers  at  the  rate  of  3  for  2  cents,  and 
sells  them  at  the  rate  of  2  for  5  cents.     How  much  does  he 
make  on  30  papers  ? 

13.  What  is  the  value    of   8    bushels    of    wheat,    if   6 
bushels  cost  $4.50  ? 

14.  What  is  the  cost  of  2  Ib.  8  oz.  of  butter  at  16  cents 
a  pound  ? 

15.  What  is  the  difference  between  5  square  feet  and  5 
feet  square  ? 

16.  When  it  is  noon  in  Syracuse,  what  time  is  it  7£°  east 
of  Syracuse  ? 

17.  Two  places  are  37|  degrees  apart.     When  it  is  5 
P.M.  at  the  eastern  place,  what  is  the  time  of  the  western  ? 

18.  When  it  is  noon  in  Syracuse,  what  is  the  time  60° 
west  of  Syracuse  ? 

19.  What  is  the  standard  time  of  Denver  when  it  is 
noon  in  Boston  ?    V 

20.  58  is  §  of  what  number  ? 

21.  A  boy  sold  a  knife  for  60  cents,  which  was  f  of  its 
cost.     What  did  it  cost  ? 


224  SENIOR    ARITHMETIC. 

22.  The  sum  of  two  numbers  is  32 ;  their  difference  is 
10.     What  are  the  numbers  ? 

NOTE.  —  The  half-sum  +  the  half-difference  =  the  greater.     The 
half-sum  —  the  half-difference  —  the  less. 

23.  At  a  village  election  there  were  1200  votes  cast  for 
two  candidates ;  the  successful  candidate  had  a  majority  of 
200  votes.     How  many  votes  were  cast  for  each  ? 

24.  The  sum  of  two  numbers  is  68 ;  their  difference  is 
26.     What  are  the  numbers  ? 

25.  What  is  33i%  of  $900  ?     66f  %  of  $1200  ?     12^% 
of  $96?     25%  of  $600? 

26.  A   merchant,   by  selling   goods   at  $80,  lost  20%. 
What  was  the  cost? 

27.  A  farmer  had  a  flock  of  sheep,  and  purchased  25% 
more ;    he  then  had   250    sheep.     How  many  had   he   at 
first  ? 

28.  A  lad  had  45  marbles,  and  lost  33^  %  of  them.    How 
many  had  he  left  ? 

29.  What  is  an  agent's  commission  for  buying  96  head 
of  cattle  at  $33£  a  head,  at  6f  %  ? 

30.  75  X  66f  -  26  X  12J  =  ? 

31.  How  much  is  500%  of  $12  ? 

32.  A  druggist  expended  $20  in  opium,  which  he  sold 
at  a  profit  of  300%.     What  did  he  sell  it  for  ? 

33.  $18  is  600%  of  what  ? 

34.  What  is  the  difference  between  .6%  of  $50  and  1% 
of  $70  ? 

35.  What  per  cent  of   a  number  is  J  of  it  ?    \  of  it  ? 
TV  of  it  ?  f  of  it  ?  f  of  it  ?  f  of  it  ?     16  is  \%  of  what  ? 

36.  A  lot  containing  48  square  rods  is  3  times  as  long  as 
it  is  wide.     What  are  its  dimensions  ? 


REVIEW.  225 

EXPLANATION.  —  As  the  length  is  three  times  the  breadth,  we 
divide  the  area  by  :>;  the  result  will  be  the  area  of  each  of  3  equal 
squares,  the  square  root  of  which  will  be  the  width,  which  multiplied 
by  3  will  give  the  length.  V^  =  4  rd.,  the  width. 

37.  A  and  B  had  the  same  income.    A  saved  1  of  his  and 
B  |.     A  had  $1600  at  the  end  of  8  years ;  how  much  had 
B? 

38.  Which  is  greater,  the  square  root  of  ^L,  or  the  cube 
of  i? 

39.  A  two-inch  pipe  can  discharge  the  contents  of  a  cask 
in  8  hours.     How  long  will  it  take  a  four-inch  pipe  ? 

40.  How  many  rods  of  fence  necessary  to  fence  a  square 
lot  containing  144  sq.  rd.  ? 

41.  A  lot  containing  144  sq.  rd.  is  four  times  as  long 
as  it  is  wide.     How  many  rods  of  fence  does  it  require  ? 
(Compare  with  result  in  Ex.  40.) 

42.  How  many  inches  in  a  hektometer  ? 

43.  How  many  milliliters  in  4  dekaliters  ? 

44.  How  many  ares  in  5  Hektares  ? 

-45.  John  and  George  divide  150  marbles  in  proportion 
to  their  ages.  John  is  7,  and  George  is  8.  How  many 
marbles  do  each  receive  ? 

46.  If  a  boy  can  ride  a  bicycle  at  the  rate  of  18  miles  an 
hour,  how  long  will  it  take  him  to  ride  twice  around  a  sec- 
tion of  land  ? 

47.  What  is  the  interest  of  $600  at  8%  for  3  months? 
for  3  years  ? 

48.  If  I  owe  a  debt  of  $60,  and  pay  $40  two  months 
before  it  is  due,  how  long  after  it   is  due  should  the  re- 
mainder be  allowed  to  run  ? 

49.  At  what  time  between  3  and  4  o'clock  are  the  hour 
and  the  minute  hand  of  a  watch  together  ? 


226  SENIOR    ARITHMETIC. 

EXPLANATION.  —  Both  hands  are  together  at  12  o'clock,  and  be- 
fore it  is  12  o'clock  again  they  will  have  been  together  11  times. 
They  will  be  together  between  1  and  2  in  ^T  of  12  hours,  and  between 
3  and  4  in  T3r  of  12  hours. 

50.  If  I  buy  8%  stock  so  that  it  pays  me  6%  on  my  in- 
vestment, what  per  cent  do  I  receive  ? 

Written. 

51.  Frost  injured  72  peach   trees  on   M's  farm,  which 
number  was  9%  of  all  the  trees  he  had.     How  many  did 
he  have  in  all  ? 

52.  At  2%  an  agent  received  $125.50  commission  on  the 
sale  of  some  real  estate.     What  was  it  sold  for  ? 


53. 

Amsterdam,  N.  Y.,  ^OH.   •/, 

after   date,    (3^  promise   to  pay 

order, 

Dollars,    with    mterest.      Value   re- 


Find  the  proceeds  of  the  above  note,  discounted  at  the 
Farmer's  National  Bank,  Amsterdam,  N.Y.,  Feb.  16,  1896. 

54.  A  gentleman   insured   his   house  for   $1800, .  which 
was  f  of  its  value,  at  1J$>.     In  case  °f  total  destruction 
by  fire,  what  is  the  entire  loss  to  the  owner  ? 

55.  A  bill  of  goods  amounting  to  $287.60  is  sold  with 
discounts  of  10^  and  5%  for  cash.     How  much  cash  will 
pay  it  ? 


KKVIKW.  H( 

56.  If  a  piano  that  cost  $360  is  to  be  sold  at  a  profit  of 
16|%,  what  price  must  be  asked  that  12-J-%  may  be  abated 
from  the  asking  price  ? 

57.  "I  sold  two  articles  for  $1.50  each,  thereby  realizing 
a  profit  of  25%    on  one  and  a  loss  of  25%  on  the  other. 
Did  I  gain  or  lose  on  both  transactions  ? 

58.  A  bought  a  carriage   at  20%    and  10%    from   list 
price,  and  sold  it  at  10%  and  5%  from  list  price.     What 
per  "cent  profit  did  he  make  ? 

59.  A  grocer  bought  a  cask  of  molasses  containing  40 
gal.  for  38  cents  per  gallon.     Seven  gallons  having  leaked 
out,  for  how  much  per  gallon  must  he  sell  the  remainder 
in  order  to  gain  12|%  on  the  investment  ? 

60.  Suppose  a  grocer  bought  a  42-gallon  cask  of  vinegar 
at  12/  per  gallon,  and  put  12  gallons  of  water  with  it,  and 
sold  it  for  the  same  price.     What  would  be  his  rate  per 
cent  gain  ? 

61.  A  meter  stick  is  what  per  cent  longer  than  a  yard 
stick  ? 

62.  Buffalo  is  the  largest  flour  depot  in  the  world.     It 
received  by  lakes  and  rail  in  1895,  8,971,740  bbls.  of  flour. 
If  the  K  Y.  C.  &  H.  K.E.  shipped  18.9%,  the  N.  Y.,  L.  E., 
&  W.   R.K  12.15%,  the  Pennsylvania  R.R.  8.33%,  the 
West  Shore  R.E.  10.97%-,  the  Lehigh  Valley  E.R.  8.12%, 
the  other  roads  6.5%,  and  the  remainder  by  water,  what 
per  cent  was  shipped  by  water  ?  and  how  many  barrels  ? 

63.  What  will  be  the  cost  of  6  loads  of  wood,  each  con- 
taining 1  C.  6  cd.  ft.  10  cu.  ft.,  at  $2.50  a  cord  ? 

64.  How  many  yards  of  carpet  2  ft.  wide  will  be  re- 
quired for  a  room  12  ft.  by  15  ft.  6  in.,  if  the  strips  run 
lengthwise,  and  there  is  a  waste  of  £  of  a  yard  in  each 
strip  in  matching  ? 


228  SENIOR,   ARITHMETIC. 

65.  The  width  of  a  building  is  36  ft.,  and  the  ridge  of 
the  roof  is  10  ft.  higher  than  the  eaves.    How  many  square 
feet  of  boards  will  it  take  to  cover  one  of  the  gable  ends  ? 

66.  With  how  long  a  rope  must  a  goat  be  fastened  to  a 
stake  that  it  may  feed  on  four  square  rods  of  land  ? 

67.  A  room  24  feet  long  and  15  feet  wide  is  to  be  car- 
peted with  carpet  |  yd.  wide.     How  many  yards  will  be 
required  if  a  waste  of  £  of  a  yard  is  made  on  each  strip  in 
matching,  the  strips  to  run  crosswise  ? 

68.  Oswego,  N.Y.,  is  in  latitude  43°  28'  N.     How  many 
degrees  is  it  from  the  North  Pole  ?    From  the  South  Pole  ? 

69.  How  many  gallons  in  32J  hektoliters  of  wine  ? 

70.  If  it  takes  2  Ib.  7  oz.  4  pwt.  of  silver  to  make  12 
spoons,  what  amount  will  be  required  for  one  spoon  ? 

71.  If  it  is  one-half  of  a  mile  from  your  home  to  the 
school  building,  how  many  steps  of  1  ft.  6  in.  each  will 
you  take  in  reaching  it  ? 

72.  What  decimal  part  of  a  week  is  4  da.  3  hr.  36  min.  ? 

73.  What  part  of  2  reams  are  10  quires,  20  sheets  ? 

74.  How  many  times  is  132  X  75  X  42  x  104  contained 
in  26  X  22  x  150  X  168  ? 

.  75.    Bought  six  loads  of  oats,  each  containing  32  bags, 
each  bag  containing  2  bushels,  worth  $.56  a  bushel,  and 
gave  in  return  8  boxes  of  ten.  each  containing  24  pounds. 
What  was  the  tea  worth  a  pound  ? 
4  x  7  X  32  X  15  X 

7b. 


16  X  56  X  5  X  4  X  6 

77.  If  f  of  a  box  of  oranges  cost  $4.50,  what  part  of  a 
box  can  be  bought  for  $5.25  ? 

78.  Simplify  the  following  complex  fraction  : 

*  X  |    .    fi 


REVIEW.  229 

79.  f  of  63  is  T72  of  what  number  ? 

80.  A  gentleman   invested  $215380   in  a  knitting-mill, 
which   was  f  of   the  value  of  the  plant.     What  was  the 

Nvalue  of  f  of  the  plant  ? 

81.  A  and  B,  being  150  miles  apart,  travel  toward  each 
other.     They  start  at  the  same  time,  and  meet  at  the  end 
of  eight  hours,  when  they  discover  that  A  has  travelled  1| 
miles  each  hour  more  than  B.     How  many  miles  has  each 
man  travelled  ? 

82.  For  how  long  a  time  must  $4560  be  placed  on  inter- 
est at  6%  to  gain  $353.40  ? 

83.  A  man  borrowed  $250  March  3,  1896,  and  paid  the 
note   Sept.  21,  1896,  with  5%    interest.      What  was  the 
amount  of  the  note  ? 

84.  A  merchant  borrowed   $165   at  6%,  and  when   he 
paid  the  debt  it  amounted  to  $168.96.     How  long  did  he 
have  the  use  of  the  money  ? 

85.  The  interest  on  a  certain  sum  is  $27.40,  the  time  2 
years,  3  months,  12  days,  and  the  rate  6%.     What  is  the 
principal  ? 

86.  A  note  for  $250  was  given  Sept.  5,  1895 ;  a  payment 
of  $75  was  made  April  25,  1896.     How  much  will  settle 
the  note  Oct.  3,  1897  ? 

87.  A  man  bought  a  farm  for  $4000,  April  1,  1889.     He 
gave   a  mortgage  at  5%  for  $3000,  and  paid  as  follows: 
Jan.  1,  1890,  $700;  Oct.   1,  1890,' $1000  ;  April  1,  1891, 
$850;   and    the    balance   of    the   mortgage  April  1,   1892. 
How  much  was  due  at  settlement  ? 

88.  What  sum  of  money  must  I  loan  at  6  per  cent  inter- 
est, that  it  may  bring  rue  in  a  quarterly  income  of  $300  ? 

89.  Compute  the  interest  on  $3450  for  2  yr.  6  mo.  20  da. 
at  6%. 


230  SENIOR    ARITHMETIC. 

90.  William  Johnson  holds  a  note  for   $1250  against 
James  W.  Way,  dated  Jan.  10,  1893,  payable  on  demand. 
This  note  bears  the   following  indorsements :   March  10, 
1893,  $200:   May  10,  1893,  $300:   July  10,  1893,  $50; 
Oct.  10,  1893,  $400.      What  is  due  Dec.  10,  1893,  interest 
at5%? 

91.  Find  the  simple  interest  of  $382.94,  one  half  to  be 
paid  in  5  yr.  5  mo.  20  days  at  3%,  the  other  half  to  be  paid 
in  5  yr.  5  mo.  20  days  at  5%. 

92.  A  man  borrows   $2000  which   belongs  to  a  minor 
who  is  18  yr.  2  mo.  10  days  old,  and  he  is  to  keep  it  until 
the  owner   is  21  years  of  age.     What  will  then   be  due, 
money  being  worth  6%  ? 

93.  Bought  a  house  for  $6000,  and  gave  a  mortgage  for 
$4000,  dated  Jan.  1,  1892,  interest  at  6%.     Made  the  fol- 
lowing payments :  July  1,  1892,  $520 ;  Jan.  1,  1893,  $708 ; 
Jan.  1,  1894,  $680 ;  July  1,  1895,  $725.     How  much  was 
due  Jan.  1,  1896  ? 

94.  A  man  owes  me  $463.50,  payable  in  6  months  with- 
out interest.     What  sum  can  I  afford  to  take  now  for  the 
debt,  money  being  worth  6%  ? 

95.  A  man  bought  goods  amounting  to  $2100  on  6  mo. 
credit,  but  was  offered  a  discount  of  3%   cash  payment. 
If  money  was  worth  £%  a  month,  what  is  the  difference? 

96.  Which  is  the  more  profitable,  to  buy  goods  worth 
$500  at  90  days,  3%  off  for  cash,  or  put  the  amount  at 
interest  at  7%,  and  let  the  bill  run  to  maturity  ? 

97.  Face  of  a  debt,  $1256.25.    Date.  July  1,  1886.    Time, 
1  yr  6  mo.     Rate,  6  % .     What  is  the  present  worth  ? 

98.  Had  a  note  of  $2500  discounted  at  a  Rochester  bank 
for  2  months.     What  were  the  proceeds,  rate  of  discount 
being  1%  ? 


REVIEW.  231 

99.  Find  the  difference  between  the  true  discount  and 
the  bank  discount  of  a  debt  of  $550,  due  in  4   months 
without  interest. 

100.  April  1,  A  gave  B  a  3-mo.  note  for  $300,  which  B 
had  discounted  at  a  bank  May  1.     What  did  B  receive  ? 
and  what  amount  could  the  bank  collect  on  July  1,  dis- 
count at  6  % ,  no  grace  ? 

101.  Bought  an  invoice  of  goods  amounting  to  $1360.58. 
How  much  will  I  make  by  discounting  my  note  at  the  bank 
for  90  days  at  6%,  and  paying  cash  for  goods  at  5%  off? 

102.  A  New  York  note  of  $2000,  bearirig  date  May  24, 
1895,  and  payable  in  60  days,  was  discounted  at  6%.     The 
discount  was  $15.     When  was  it  discounted  ? 

103.  Sweet  and  Johonnot  sold  20  bicycles  to  a  dealer, 
taking  his  note  at  60  days,  which  they  discounted  immedi- 
ately at  the  Merchant's  Bank  at  6%  with  grace,  receiving 
$1485.     What  was  the  price  of  each  bicycle  ? 

104.  On  the  first  day  of  January,  1890,  a  man  gave  three 
notes,  the  first  for  $500  payable  in  30  days ;  the  second  for 
$400  payable  in  60  days;  and  the  third  for  $600  payable 
in  90  days.     WThat  was  the   average  term  of  credit,  and 
what  the  equated  time  of  payment  ? 

105.  I  wish  to  use  $560.88  immediately.     For  what  sum 
must  I  draw  a  bank  note,  due  in  96  days  at  6%,  that  I  may 
receive  the  required  amount  ? 

106.  How  many  $500    U.   S.   bonds   can   be  bought  for 
$6630  at  10L%  premium? 

107.  A  guardian  invests  $1000  at  simple  interest  at  3%, 
$1000  in  4%  bonds  at  112i,  and  $1000  in  5%  bonds  at 
125.     The  bonds  are  to  run  10  years,  and  be  redeemed  at 
par.     Compare  the  three  investments  at  the  end  of  the  ten 
years. 


232  SENIOR    ARITHMETIC. 

108.  The  city  of  Buffalo  pays  $12425.72  for  rented  school 
buildings.     On  what  amount  of  3%c/0  bonds  would  this  pay 
the  interest  ? 

109.  If  I  buy  bank  stock  at  20%  discount,  and  sell  it  at 
10%  premium,  what  per  cent  do  I  gain  ? 

110.  What  is  the  rate  of  income  on  a  4%  stock  bought 
at  62J. 

111.  I  have  $5000  to  invest,  and  can  buy  5%  stock  at 
110,  or  6%  stock  at  125.     Which  will  be  the  better  invest- 
ment ?  and  how  much  annually  ? 

112.  A  gentleman  owned  a  house  which  he  rented  for 
$375  above  all  expenses.     He  sold  the  house  for  $5000, 
and  invested  the  money  in  a  5%  stock  at  80.     Did  he  gain 
or  lose  by  the  transaction  ?  and  how  much  per  year  ? 

113.  The  ratio  of  A's  weight  to  that  of  B  is  |.    B  weighs 
120  Ib.  8  oz.     What  does  A  weigh  ? 

114.  If  John  is  6  years  old  and  Henry  15,  what  is  the 
ratio  of  John's  age  to  that  of  Henry  ?     What  will  it  be 
when  each  is  5  years  older  ? 

115.  If  4  horses  eat  4  bu.  of  oats  in  2  days,  how  many 
horses  will  eat  48  bushels  in  12  days  ?    (Solve  by  analysis.) 

116.  If  the  antecedent  is  §  of  T9g  x  ?^j,  and  the  ratio  is 
§  of  29?>  what  is  the  consequent? 

117.  How  wide  can  20  men,  working  8  hours  a  day  for 
8  days,  make  a  ditch  which  is  75  rods  long  and  10  ft.  deep, 
if  25  men,  working  10  hours  a  day  for  7  days,  can  dig  a 
ditch  80  rd.  long,  8  ft,,  deep,  and  2  ft.  wide  ? 

118.  If  a  baker's  loaf  weighs  10  ounces  when  wheat  is 
60  cents  a  bushel,  what  should  it  weigh  when  wheat  is  70 
cents  a  bushel  ? 

119.  If  a  train  moves  at  the  rate  of  30  miles  in  48  min- 
utes, in  what  time  will  it  run  450  miles? 


REVIEW.  233 

120.  One  side  of  a  shed  is  8  ft.  high,  the  opposite  side  13 
ft.  6  in.     What  is  the  ratio  between  the  sides  ? 

121.  If  it  costs  $30  to  lay  a  cement  sidewalk  4  ft.  wide 
and  16  ft.  long,  how  much  will  it  cost  to  lay  the  same 
kind  of  walk  7  ft.  wide  and  96£  ft.  long  at  the  same  rate  ? 

122.  Write  and  solve  a  problem  in  proportion,  using  the 
following  numbers:  8  men,  9  Ib.  of  beef,  1  da. ;  and  2  da., 
12  Ib.  of  beef. 

123.  If  T\  of  a  yard  of  cloth  cost  $1,  what  will  4£  yd. 
cost? 

124.  A,  B,  and  C,  engaged  in  trade.     A  put  in  $400,  B 
$250,  C  $600  j  they  gain  $300.     Find  each  man's  share 
of  the  gain. 

125.  A  merchant  failing  in  trade  has  debts  amounting  to 
$34560 ;  his  assets  are  $30240.     What  can  he  pay  on  the 
dollar  ?  and  how  much  will  a  creditor  receive  to  whom  he 
owes  $3840  ? 

126.  A  man  willed   his  property,  which  was  valued  at 
$6000,  to  his  four  children  in  the  following  proportion, 
giving  to  each  one  £,  ±,  £,  and  £  respectively.     How  much 
did  each  one  receive  ? 

127.  Three  families  rent  a  cottage  for  the  summer.     The 
first  family  occupies  it  for  6  weeks,  the  second  for  2,  and 
the  third  for  3  weeks.     The  rent  for  the  entire  season  of 
11  weeks,  is  $440.     How  much  should  each  family  pay  ? 

128.  Scrantom,  Morris,  and  Jackson  were  associated  in 
business  for  a  period  of  1  yr.  6  mo.     Scrantom  furnished 
$5000,  Morris  $3000,  and  Jackson  $2000  of  the  original 
capital.     When  the  partnership  terminated,  they  divided 
$4000,  the  profits  arising  from  the  same.     How  much  more 
did  each  make  than  he  would  have  realized  had  his  money 
been  invested  in  a  6%  mortgage  ? 


234  SENIOR    ARITHMETIC. 

129.  Divide  $60  among  three  boys  so  that  one  shall  have 
£  as  much  as  the  other  two,  whose  shares  are  as  3  to  7. 

130.  What  is  the  distance  between  the  diagonally  oppo- 
site corners  of  a  lot  whose  area  is  16  sq.  ft.  ? 

131.  My  dining-room  is  16  ft.  long,  14  ft.  wide,  10  ft. 
high.     Find  diagonals  of  the  shorter  sides,  of  the  longer 
sides,  and  of  the  room. 

132.  What  is  the  length  of  one  side  of  a  cube,  equal  in 
volume  to  a  solid  that  is  49  ft.  long,  27  ft.  wide,  and  7  ft. 
high? 

133.  A  ladder  25  ft.  long,  the  bottom  of  which  is  5  ft. 
from  a   building,   reaches    the  base  of   a  window.     How 
many  feet  from  the  base  of  the  window  to  the  ground  ? 

134.  At  40  cents  a  rod  for  fencing,  which  will  cost  the 
more,  to  enclose  a  square  field  containing  10  A.,  or  a  field 
of  the  same  area  whose  length  is  twice  its  width  ? 

135.  Find  the  cube  root  of  41781.923. 

136.  Find  the  square  root  of  41781923. 

137.  A  cubical  cistern  contains  30  hhd.  of  water.     How 
deep  is  it  ? 

138.  The  volume  of  a  rectangular  prism  is  200  cu.  ft., 
and  its  height  is  8   ft.     Find  its  surface  contents,  if  its 
two  other  dimensions  are  equal. 

139.  The  area  of  a  right-angled  triangle  is  289  sq.  ft.,  its 
base  is  £  of  its  altitude.     What  is  the  length  of  its  alti- 
tude ? 

140.  If  a  railroad  company  pays  19/  per  sq.  yd.  for  ex- 
cavating, and  37£/  per  sq.  yd.  for  drawing  away  the  earth, 
what  will  it  cost  the  company  to  remove  a  mound  equal 
in  volume  to  a  cube  whose  side  is  81  feet  ? 


REVIEW.  235 

141.  Forty  feet  directly  east  from  a  column  that  is  75 
ft.  high,  I  measure  clue  north  30  ft.,  and  find  that  I  am  in 
line  with  a  stake  and  the  column.     If  the  stake  is  25  ft. 
distant  from  my  position,  and  10  ft.  high,  what  is  the  dis- 
tance from  the  top  of  the  stake  to  the  top  of  the  column  ? 

142.  How  many  rods  of  fence  will  enclose  a  rectangular 
field  containing  20  acres,  if  the  field  is  twice  as  long  as 
it  is  wide  ?  and  how  much  will  it  cost  at  $2.45  per  rod  ? 

143.  If  a  locomotive  runs  at  the  rate  of  55  miles  in  40 
minutes,  and  its  drive-wheels  are  18  ft.  in  circumference, 
how  many  revolutions  will  the  drive-wheel  make   in  one 
hour  ? 

144.  A  insured  his  stock  for  $1200.     He  paid  a  premium 
of  $24.     What  was  the  rate  of  insurance  ? 

145.  Grant  and  Dunn  bought  a  bill  of  glass  amounting 
to  $853.68,  upon  which  they  received  a  discount  of  60%, 
25%,   15%,    and  2%   off   for  cash.      What  was   the  net 
amount  of  bill  ? 

146.  My  agent  in  Chicago  sold  goods  to  the  amount  of 
$8640.     He  also  purchased  6800  bu.  of  wheat  at  $1.10  a 
bushel,  paid  for  expenses  $10.40,  and  received  a  commis- 
sion of  2  ct.  on  every  dollar.     How  much  will  he  remit 
to  me  after  paying  all  expenses  ? 

147.  A  merchant  buys  calico  at  5|   ct.   per  yard,   and 
sells  at  6.     What  is  his  rate  per  cent  of  gain  ? 

148.  What  is  the  rate  of  insurance  when  a  $1000  policy 
for  3  years  costs  $7.50  ? 

149.  A  man  lost  $13.45  on  some  flour  by  selling  it  at  a 
loss  of  14f  %.     What  was  the  flour  worth? 

150.  A  farmer  buys  4  tons  of  hay  at  $20  per  ton,  and 
4  bbl.  of  flour  at  $5  per  barrel.     What  is  the  cash  value 
of  the  bill,  if  he  is  allowed  a  discount  of  15%,  and  5% 
deduction  for  cash  ? 


236  SENIOR    ARITHMETIC. 


TEST   QUESTIONS. 

374.  Arranged,  by  permission,  from  examinations  given 
in  various  cities. 

1.  Define  addition,  sum,  sign  of  equality,  subtraction, 
remainder,  subtrahend,   minuend,   parenthesis,   multiplica- 
tion, factors,  multiplicand. 

2.  Multiply  7258  by  395,  and  write  each  partial  pro- 
duct in  words. 

3.  Subtract  8969  from  9782,  and  prove  the  work. 

4.  Prove    by   an   illustration    that    multiplication    re- 
sembles addition. 

5.  Solve  :  $73.46  -  ($.94  +  $3.02)  +  $47  X  35. 

6.  Write  in  figures,  XL VI I. 

7.  Write  in  figures,  six  hundred  eight  thousand  seventy- 
two. 

8.  Multiply  6504  by  657. 

9.  I  of  585  x  5  =  ? 

10.    Divide  45897  by  490,  and  prove  that  your  work  is 
correct. 

375.  1.    Copy   and   find    the    sum:    $23.17,    $6043.05, 

$0.42,  $208.97,  $5486.04. 

2.  How  many  yards  of  linen,  at  28  cents  a  yard,  must 
be    given   for   35    bushels    of   potatoes,    at   56   cents   per 
bushel  ? 

3.  A  man  paid  $13,465  for  a  house  and  some  land. 
The  house  alone  was  worth  $8,978.     What  was  the  value 
of  the  land  ? 

4.  Write  this  number  in  words,  3,782,013. 

5.  Write  in  words,  XCV.  ;  76508.904. 


TEST   QUESTIONS.  237 

6.  How  many  bushels  of  potatoes  at  50  cents  a  bushel 
will  pay  the  entire  cost  of  a  hat  at  $7.50,  a  dress  at  $24, 
a  cloak  at  $16.25,  and  gloves  at  $1.75  ? 

7.  6460000  x  3000  -v-  25000000  =  ?      (Use    shortest 
way.) 

8.  If  the  dividend  is  1761184  and  the  quotient  4684, 
what  is  the  divisor  ? 

9.  What  is  the  smallest  number  that  will  exactly  con- 
tain 16,  20,  24,  or  30  ? 

10.  Define  dividend,  remainder,  product,  the  prime  fac- 
tors of  a  number.  How  do  you  prove  division  ? 

376.    1.    Define  multiplier ;  concrete  number. 

2.  How  can  you  prove  an  example  in  subtraction  ? 

3.  A  merchant  bought  375  bbl.  of  apples  at  $.95  a  bbl. ; 
43  bbl.  rotted ;  if  he  sells  the  rest  at  $1.10  per  bbl.,  how 
much  does  he  gain  or  lose  on  all  ? 

4.  My  salary  is  $2350  a  year,  and  I  spend  $4  a  day ; 
how  much  will  t  save  in  six  years  ? 

5.  If  23  men  own  475  bbl.  of  apples  each,  and  4  of 
them  divide  theirs  equally  among  the  rest,  how  many  will 
each  have  then  ? 

6.  Find  the  prime  factors  of  1155. 

7.  If  my  salary  is  $1400  per  year,  and  my  expenses 
$90  per  month,  how  long  will  it  take  me  to  save  $4160  ? 

8.  What  is  the  smallest  quantity  of  grain  that  will  fill 
an  exact  number  of  bins,  whether  they  hold  312,  260,  or 
390  bushels  ? 

9.  What  are  like  numbers  ?     Give  three. 

10.  From  Albany  to  West  Troy  is  5  miles,  from  West 
Troy  to  Cohoes  2  miles,  and  from  Cohoes  to  Saratoga  is  30 
miles.  How  far  is  it  from  Albany  to  Saratoga  ? 


238  SENIOR    ARITHMETIC. 

377.  1.    Write  in  words  23456789. 

2.  Find  the  greatest  common  divisor  of  75,  25,  and  500, 
and  their  least  common  multiple. 

3.  If  7  tons  of  hay  cost  $105,  what  will  be  the  cost  of 
289  tons  ? 

4.  Write  the  number  which  is  composed  of  3  units  of 
the  eighth  order,  6  of  the  fifth,  2  of  the  third,  and  9  of  the 
second. 

5.  Find  the  contents  of  the  smallest  measure  that  may 
be  filled  by  using  either  a  4-quart,  a  5-quart,  or  a  6-quart 
measure. 

6.  Find  the  prime  factors  of  1452. 

7.  Solve  by  cancellation  :  A  man  receives  $21  for  15 
days'  work  of  7  hours  each.     How  much  should  he  receive 
for  19  days'  work  of  5  hours  each  ? 

8.  The  product  of  three  numbers  is  105840 ;  one  of  the 
numbers  is  42,  the  other  35.     What  is  the  third  number  ? 

9.  How   many   pounds  of  butter  at  20/  a  pound  are 
worth  as  much  as  1600  bushels  of  wheat  at  75/  a  bushel  ? 

10.    What  is   the  greatest    common   divisor   of   two   or 
more  numbers  ? 

378.  1.    Two  persons   start   from  the    same    point   and 
travel  in  opposite  directions  ;  one  at  the  rate"  of  25  miles 
a  day,  and  the  other  at  the  rate  of  32  miles  a  day.     How 
far  apart  will  they  be  in  8  days  ? 

2.  •  What  is  the  product  of  20202  x  10101  ? 

3.  What    number    multiplied    by    1728    will    produce 
1705536  ? 

4.  A  man  has  $8250  ;  how  much  must  he  add  to  this 
to  be  able  to  pay  for  a  farm  worth  $10000  ? 


TEST   QUESTIONS.  239 

5.  (6070  -  1200)  +  (4680  -=-  15)  =  ? 

6.  Bought  144  acres  of  land  at  $41.25  an  acre,  and  sold 
the  whole  for  $7000.    Did  I  gain  or  lose  ?  and  how  much  ? 

7.  If  3  oranges  are  worth  f  of  a  melon,  what  part  of 
the  melon  is  1  orange  worth  ? 

8.  Austin  having  30  marbles,  gave  £  of  them  to  one 
companion  and  I  of  them  to  another.     How  many  had  he 
left? 

9.  How  many  eggs  in  12^  dozen  ? 

10.    Write  the  present  year  in  Roman  numerals. 

379.  1.    George  gave  a  beggar  9  cents,  which  was  £  of 
all  the  money  he  had.     How  much  money  had  he  ? 

2.  Mary  is  14   years  old,  and  her  sister  is  f  as   old. 
How  old  is  her  sister  ? 

3.  What  is  a  mixed  number  ? 

4.  How  many  ninths  in  5£  ? 

5.  At  |  of  a  dollar  a  pound,  what  will  8  pounds  of 
butter  cost  ? 

6.  What  will  |  of  a  pound  of  coffee  cost  at  28  cents 
a  pound  ? 

7.  If  a  man  earns  $15  a  week  and  spends  f  of  it,  how 
much  does  he  save  ? 

8.  What  do  you  understand  by  |  of  anything  ? 

9.  Change  142 1  to  an  improper  fraction. 

10.    A  boy  having  20  quarts  of  blueberries,  sold  f  of  them 
for  $£%.     What  was  the  price  for  a  quart  ? 

380.  1.    If  I  put  £  of  my  money  in  one  bank,  \  in  an- 
other, £  in  another,  and  have  $4,200  besides,  how  much 
have  I? 


240  SENIOR   ARITHMETIC. 

2.  A  can  mow  a  field  in  10  days,  B  in  8  days,  and  C  in 
5  days.     When  working  together,  how  many  days  will  they 
need? 

3.  If  6  is  added  to  both  terms  of  the  fraction  -J,  how 
much  is  the  fraction  increased  or  diminished  ? 

4.  The  divisor  is  46,  the  quotient  605,  and  the  remain- 
der 23.     What  is  the  dividend  ? 

5.  From  the  sum  of  |  and  |  take  the  sum  of  T\  and  §. 

6.  How  many  cords  of  pine  wood  at  $3.25  a  cord  must 
be  given  for  12  yards  of  broadcloth  at  $2.10  a  yard  ?  Work 
and  analyze. 

7.  Find  the  prime  factors  of  13860. 


8. 


13  x  16  X  42  x  51  _  9 


6  X  17  x  48  x  91 
9.    Find  the  sum  of  the  prime  numbers  under  20. 
10.    Reduce  |f|^  to  lowest  terms. 

381.    1.    Write  in  Roman  notation  1894. 

2.  Write  the  prime  numbers  from  1  to  18  inclusive. 

3.  If  3  boxes  of  oranges  cost  $5f,  how  many  boxes  can 
be  bought  for  $17  ? 

4.  A  farmer  sold  64  sheep,  and  had  ^  of  his  flock  left. 
How  many  had  he  left  ? 

of  3' 


6.  How   many   barrels   of    flour  at   $6   a   barrel   must 
be  given  for  3   pieces  of  linen,  each  containing  36  yds., 
at  25  ct.  a  yard  ? 

7.  A  farmer  sold  at  market  15  sheep  at  $-^  each,  and 
bought  7  yards  of  cloth  at  $1|-   per  yard.      How  much 
money  did  he  take  home  ? 


TEST    QUESTIONS^pg^/ 

8.  From  the  sum  of  5$,  9£,  11$,  take  the  difference 
between  32  and  13 1. 

9.  Reduce  to  their  least  common  denominator  Jj,  ||, 

21       24 

2S>   §S* 

10.    Write  a  receipt    for    $20    paid    you   by  Mr.   John 
Dixon. 

382.    1.    Add  $49.50,  $43.62$,  $75.05,  $64.75,  $35.09, 
$6.03$,  $42,  $73.98,  $105.60. 

2.  How  many  sheep  at  $5  each  must  be  given  for  15 
horses  at  $150  each  ? 

3.  What  is  the  sum  of  ^i  -f  ^  +  t  ? 

6          o        4 

4.  From  §  of  J  of  3  take  £  of  1J. 

5.  At  $39 1  apiece,  how  many  cows  can  be  bought  for 


6.  How  many  times  is  T30  of  §  of  6|  contained  in  |  of 
54  X  §  -*-  J  ? 

7.  Define  multiple  and  greatest  common  divisor. 

8.  Give  and  define  proper  fraction  ;  mixed  number. 

9.  If  a  man  spends  £  of  his  money  for  a  house,  f  for  a 
farm,  and  has  $3400  in  cash  left,  what  is  the  amount  of 
his  wealth  ? 

10.  A  grocer  bought  3  barrels  of  apples  of  different 
qualities  at  $2.75,  $3.12,  and  $3.25  a  barrel.  What  was 
the  average  cost  ? 

383.    1.    What  is  reduction  of  fractions  ? 

2.  Reduce  }J  to  156ths. 

3.  Express  T27\\  in  its  simplest  form. 

4.  Change  to  fractions  having  the  least  common  denom- 
inator, J,  T*j,  and  TV 


242  SENIOR   ARITHMETIC. 

5.  If  a  merchant  buys  tea  at  $f  a  pound,  and  sells  it 
at  $|?  does  he  gain  or  lose  ?  and  how  much  ? 

6.  Find  the  sum  of  f ,  7£,  and  8j. 

7-  A  +  A  +  Wk  +  i-*? 

8.   A  man  engaged  to  labor  30  days,  but  was  absent  5T7^ 
days ;  how  many  days  did  he  work  ? 

10.  A  young  man  received  a  salary  of  $60 f  a  month, 
and  paid  for  his  board  $30£,  for  washing  $lj,  and  for 
other  expenses  $12T9S.  How  many  dollars  had  he  left  ? 

384.  1.    Define  a  proper  fraction,  and  give  an  example 
of  one. 

2.  A  merchant  bought  three  pieces  of  cloth  containing 
125$,  96|,  and  48f  yards.     How  many  yards  did  he  buy  ? 

3.  What  is  the  value  of  2£  times  |  of  f  of  l£  ? 

4.  If  9  men  consume  f  of  9|  pounds  of  meat  in  a  day, 
how  much  does  one  man  consume  ? 

5.  A  farmer  distributed  15  bushels  of  corn  among  sev- 
eral persons,  giving  them  1|  bushels  apiece.     Among  how 
many  persons  did  he  divide  it  ? 

6.  What  is  the  value  of  lii  ? 

7.  What  number  must  be  added  to  22 §  that  the  sum 
may  be  99^  ? 

8.  A  can  do  a  piece  of  work  in  8  days,  and  B  can  do  it 
in  6  days.     In  what  time  can  they  do  it  working  together  ? 

9.  A  pole  stands  ^  in  the  mud,  J  in  the  water,  and  21 
feet  above  the  water.     What  is  its  length  ? 

10.  A  man  bequeathed  to  his  son  $3500,  which  was  f  of 
what  he  left  his  wife.  How  much  did  he  leave  his  wife  ? 

385.  l.    If  §  of  a  farm  is  valued  at  $1728,  what  is  the 
value  of  the  whole  ? 


TEST    QUESTIONS.  243 

2.  If  8  be  added  to  both  terms  of  the  fraction  f ,  will 
its  value  be  increased  or  diminished  ?  and  how  much  ? 

3.  If  the  sum  of  two  fractions  is  f ,  and  one  of  them  is 
•fjfj  what  is  the  other  ? 

4.  Express  in  its  simplest  form  the  quotient  of  2025 
divided  by  3645. 

5.  If  the  dividend  is  |  and  the  quotient  3\,  what  is  the 
divisor  ? 

6.  At  $J  a  bushel,  how  many  bushels  of  apples  can  be 
bought  for  $5 1  ?     (Analysis.) 

7.  Define  fraction,  terms  of  a  fraction,  improper  frac- 
tion, compound  fraction,  and  complex  fraction. 

8.  Change  5T2g*  to  a  whole  or  mixed  number. 

9.  How  many  8ths  of  a  bushel  in  9£  bushels  ? 

10.  Change  g,  |,  T75,  T52,  f  to  equivalent  fractions  having 
a  common  denominator. 

386.  1.  A  farmer  sells  6  jars  of  butter  holding  8  pounds, 
at  36/  a  pound,  and  receives  in  payment  14  cans  of  coffee, 
each  holding  two  pounds.  What  was  the  price  of  the  cof- 
fee ?  Work  by  using  cancellation. 

2.  If  a  man  walks  3^  miles  in  one  hour,  how  far  can  he 
walk  in  9  hours  ? 

3.  Find  the  sum  of  13§,  }£,  6f,  20|f,  and  17H. 

4.  How  many  days'  work  at  $lf  a  day  will  pay  for  8\$ 
yards  of  cloth  at  $2^  a  yard,  and  56  Ib.  of  butter  at  25 
cents  a  pound  ? 

5.  The  product  of  two  numbers  is  41|,  and  one  of  them 
is  160Tf  j  ;  what  is  the  other  ? 

6.  Find  the  sum  of  93567  +  20754867  +  4756  +  925674 
+  6543987  +  6579  +  98675  +  567923  +  645876  +  9346  + 
878  -f  54562  +  888. 


244  SENIOR   ARITHMETIC. 

7.  Tell  in  words  what  these  numbers  are :  1950 ;  90 ; 
4040;  73000007. 

8.  Find  the  difference  between  76392  X  4506  and  985301 
X  976. 

9.  What  will  79  ten-ton  cars  of  coal  be  worth  at  $5.50 
a  ton? 

10.  If  you  should  buy  376  horses  for  $65123,  how  much 
would  you  sell  them  for  apiece  to  gain  $5189  ? 

DECIMALS. 

387.  1.    A  merchant  bought  four  pieces  of  cloth  contain- 
ing 32f,  38 1,  40|,  45|    yards,  respectively.      How  many 
yards  did  he  buy  ? 

2.  Change  to  decimals  and  add  :  |,  f ,  4|. 

3.  From  a  farm  containing  128 f  acres,  84|  acres  were 
sold.     How  many  were  left  ? 

4.  Find  the  prime  factors  of  1008. 

5.  50  -5-  .05  =  ? 

6.  What  will  2.47  pounds  of  coffee  cost  at  $.48   per 
pound  ? 

7.  If  one  yard  of  ribbon  costs  34i  cents,  what  will  6 
pieces  cost,  each  piece  containing  13.12  yards? 

8.  Write  in  words  68.0642. 

9.  If  9  yards  of  cloth  cost  $1.17,  what  will  15  yards 
cost? 

10.  A  lady  went  shopping  with  $45.  She  paid  $4-^ 
for  shoes,  $5f  for  a  hat,  $12|  for  a  dress.  How  much 
money  had  she  left  ? 

388.  1.    If  a  farm  is  worth  $3200,  how  much  is  |  of  it 
worth  ? 

0 

2.    From  one  million  take  one  millionth. 


TEST   QUESTIONS.  245 

3.  What  is  the  difference  in  cents  between  |  of  a  dollar 
and  |  of  a  dollar  ? 

4.  Point  off  into  periods  96308796,  and  write  over  each 
period  its  name. 

5.  Express  with  figures  the  following  numbers  :  Seven 
million  ninety-five  thousand,  sixty-three  and  fifteen  thou- 
sandths, and  seven  hundred  and  seven  hundredths. 

6.  Head    (write   in   words)    the   following  :  642.0016  ; 
100.01. 

7.  353812416  --  589  =  ? 

8.  Find  the  sum  of  684.8,  96.84,  6.0T5,  .1906,  7508. 

9.  At  $9  per  M.,  what  will  6728  feet  of  lumber  cost  ? 
10.    At  $.65  per  C.,  what  will  1240  pens  cost  ? 

389.    1.    Eeduce  to  a  simple  fraction        ° 


2.  What  fraction  of  11  f  is  5|  ?  ° 

3.  What  common  fraction  equals  .0125. 

4.  Reduce  $T3^  to  a  decimal. 

5.  A  farmer  sold  120  sheep,  which  were  f  of  his  flock. 
How  many  had  he  before  the  sale  ? 

6.  A  grocer  sold  ^  of  a  barrel  of  sugar  to  one  man  and 
i  of  it  to  another,  and  had  80  pounds  left.     How  many 
pounds  did  the  barrel  contain  at  first  ? 

7.  A  and  B  can  do  a  piece  of  work  in  12  days,  A  can 
do  it  in  25  days  ;  in  how  many  days  can  B  do  it  ? 

8.  A  man  spent  f  of  his  money  for  a  horse  and  f  of  the 
remainder  for  a  buggy  and  harness,  and  had  $37.50  left. 
How  much  money  had  he  at  first  ? 

9.  In  dividing  by  a  decimal,  how  do  you  determine  the 
proper  place  of  the  decimal  point  in  the  quotient  ? 

10.    At  $9.75  per  thousand,  what  will  16544  bricks  cost  ? 


240  SENIOR    ARITHMETIC. 

390.  1.  Which  is  the  greater,  £J  or  f  g  ?  How  much 
greater  ? 

2.  Find  the  sum  of  78f ,  87T\,  4f,  and  79f . 

3.  A  man  has  three  lots,  which  are  120,  420,  and  600  ft. 
wide  respectively.     He  wishes  to  divide  them  into  lots  oi 
the  greatest  equal  width  possible.     How  wide  will  each  lot 
be  ?     How  many  such  lots  can  he  make  ? 

4.  If  .375  of  a  ton  of  coal  cost  $2.40,  what  is  the  price 
of  one  ton  ?     How  many  tons  can  be  bought  for  $80  ? 

5.  Reduce  to  decimals  f ,  f,  f,  J£,  y1^. 

6.  Write  in  figures  thirteen  thousandths,  four  hundred 
and  five  hundredths,  five  hundred  fifteen  millionths,  and 
add  the  results. 

7.  Eeduce  to  a  simple  fraction  Tg  °    T?  . 

f +  1 

8.  Sold  a  house  for  $4,797,  which  was  two-sevenths 
more  than  it  cost ;  find  the  cost  price. 

9.  Make  a  bill  for  the  following  articles,  bought  to-day 
of  James  Brown,  No.  23  Warburton  Avenue,  Youkers,  N. 
Y.  :  30  oranges  at  25  cents  a  dozen ;  7  Ib.  of  coffee  at  28/ ; 
3 1  Ib.  prunes  at  13/ ;  1  bag  of  sugar  containing  28  Ib.  at 
5^/.     Receipt  the  bill  as  though  you  were  James  Brown's 
clerk. 

10.  Bought  three  boxes  of  oranges  containing  263,  220, 
and  156,  at  $3.50  per  hundred,  and  sold  them  for  50^  per 
doz.  Find  the  amount  of  profit. 

391     i     (7|  -  2.05)  -v-  (5  X  .23) 
.7|  +  2.23£  -  .6  -j-  .4 

2.  1000  -^-  .001  =  ? 

3.  j  +  i  +  .75  +  1|  +  -330  =  ? 

4.  Reduce   7    to  a  decimal. 


TEST  QUESTIONS.  247 

6-    T7<y  +  18*  -  T*J<F  +  A  +  A  =  ? 

6.  J  X  10.0019  X  1.2  X  j  X  .463  =  ? 

7.  Change  .0507  to  hundredths. 

8.  Change  8.84  to  a  common  fraction    in   its    lowest 
terms. 

9.  .123  -  .01  -  .11  -  .003  =  ? 
10.    .0509  +  I  -  .27  =  ? 


392.  1.    (.05015  -f-  2.006)  +  (24.6  -=-  .0012  x  TV)  - 
1200f  =  ? 

2.  Express  in  words  the   following:  10020.00042024; 
.000702 ;  .00000018  ;  30000.00030  ;  .00010020. 

3.  If  a  man  travels  at  the  rate  of  7.4  miles  an  hour, 
how  long  will  it  require  to  travel  370  miles  ? 

4.  What  will  be  the  cost  of  3|  yd.  of  cloth  at  .75  dollars 
per  yard  ? 

5.  Find  the  sum  of  .125,  46.42,  9.3,  164.25,  .80406. 

6.  From  1000  subtract  .001. 

7. '  Find  the  cost  of  445.375  bushels  of  wheat  at  $.9173 
per  bushel. 

8.  Change  .00125  to  a  common  fraction. 

9.  Reduce  T4^  to  a  decimal. 

10.    At  $.044  per  pound,  how  many  pounds  of  sugar  can 
be  bought  for  $44  ? 

393.  1.    Find  the  cost  of  9^  tons  of  coal,  if  |  of  a  ton 
cost  $3.00. 

2.  Find  the  sum  of  40  units,  20  tens,  464  thousandths, 
5  ten-thousandths,  and  1  millionth. 

3.  Write   in  figures  two  and  twenty-six  hundredths  ; 
two  and  twenty  six-hundredths. 


248  SENIOR    ARITHMETIC. 

4.  What  number  multiplied  by  14|  will  produce  1684^  ? 

5.  If  f  of  a  yacht  is  valued  at  $3840^,  what  is  the  value 
of  the  whole  ? 

6.  If  f  of  a  pound   of  tea  cost   $.50,  what  will   16J 
pounds  cost? 

7.  Reduce  to  simplest  form: 


1  of      x  - 

6£       llf 

8.  Reduce  }i|  to  a  decimal. 

9.  A  man  bequeathed  T7^  of  his  estate  to  his  elder  son, 
and  the  remainder  to  his  younger  son,  who  received  $1344. 
What  was  the  estate  worth  ? 

10.    What  must  be  paid  for  8960  pounds  of  plaster  at 
$5.50  per  ton  ? 

DENOMINATE   NUMBERS. 

394.    1.    Define  simple  quantity  ;  compound  quantity. 

2.  Reduce  1760  cwt.  to  higher  denominations. 

3.  Add:  11  oz.  11  pwt.  15  gr.  ;  7  oz.  12  pwt.  19  gr.  ; 
10  oz.  13  pwt.  17  gr. 

4.  Write  the  table  of  long  measure. 

5.  Find  the  total  area  in  sq.  yards  of  the  ceiling  of  a 
room  18  ft.  long,  and  15  ft.  wide. 

6.  Find  the  number  of  square  feet  in  the  surface  of  a 
cube  3  ft.  by  3  ft.  by  3  ft. 

7.  Find  the  total  area  in  the  four  walls  of  a  room  18  ft. 
long,  15  ft.  wide,  and  9  ft.  high. 

8.  Define  fraction  ;   mixed  number  ;    proper  fractions  j 
improper  fractions. 

9.  Reduce  |,  f  ,  and  /?  to  similar  fractions. 
10.    Define  circumference  ;  diameter. 


TEST   QUESTIONS.  249 

395.  1.    What  is  the  value  of  §  of  3  divided  by  £-  of  f 
plus  J  of  |  ? 

2.  A  and  B  can  build  a  shop,  working  together,  in  10 
days  ;   B  can  build  it,  working  alone,  in  30.     In  how  many 
days  can  A  build  it  ? 

3.  Add  0.525  mi.,  0.125  rd.,  0.5  yd.,  and  0.16  ft. 

4.  From  T2r  of  a  square  rod  take  |  of  a  square  yard. 

5.  Find  »  of  9  A.  70  sq.  rd.  15  sq.  yd.  7  sq.  ft.  19  sq.  in. 

6.  There  is  a  room  15  ft.  long,  12  ft.  wide,  and  9  ft. 
high ;  it  has  2  windows,  each  3  ft.  by  6  ft.,  and  a  door  3  ft. 
by  7  ft.     Taking  out  the  space  for  door  and  windows,  how 
much  will  it  cost  to  plaster  this  room  at  25/  per  square 
yard  ? 

And  what  will  be  the  cost  of  floor  boards  l£  in.  thick,  to 
lay  the  floor  of  this  room  at  $40  per  thousand  ? 

7.  There  is  a  square  field  40  chains  around ;  how  many 
acres  are  in  it  ? 

8.  In  a  space  27  ft.  long,  18  ft.  wide,  and  12  ft.  high, 
there  may  be  placed  how  many  cubes  3  feet  pn  each  edge  ? 

9.  How  many  grains  in  a  ton  ?     How  many  gallons  in 
a  cu.  yard  ? 

10.    How  many  grains  in  a  Troy  pound  ? 

396.  1.    What  decimal  of  a  mile  is  \  of  5  mi.  89  rd.  3  yd. 
2  ft.? 

2.  Divide  15  T.  17  cwt.  29  Ib.  7  oz.  by  f . 

3.  36 £  sq.  in.  equals  what  fraction  of  an  acre  ? 

4.  I  mi.  +  |  rd.  +  1  ft.  -  7i  yd.  =  ? 

5    If  7  spoons  weigh  7  oz.  12  pwt.  9  gr.,  what  will  13 
similar  spoons  weigh  ? 

6.    Add  36 J,  .00125,  1460,  |,  T%,  and  16.26. 


250  SENIOR    ARITHMETIC. 

7.  If  |  of  a  ship  is  worth  $6285,  what  is  T5^  worth  ? 

8.  To-day  you,  as  a  clerk  of   Chester  &  Wilson,  sell 
Win.  Lambert  20  bbl.  flour  at  $4.87£,  4500  Ib.  meal  at 
$1.06  per  cwt,  arid  2450  Ib.  bran  at  $13.50  per  T.     Make 
out  the  proper  bill. 

9.  Define  improper  fraction ;   decimals ;    reduction  de- 
scending ;  a  bill. 

10.    Reduce  £17  14s.  3far.  to  farthings,  and  prove. 

397.  1.  For  22  Ib.  14  oz.  of  butter  worth  16/  a  pound 
a  man  gets  12  quarts  of  sirup.  What  is  the  price  of  the 
sirup  per  gallon  ? 

2.  At  $3  a  perch,  what  would  masons  earn  in  laying  a 
wall  8  ft.  high  and  2  ft  thick  in  a  cellar  dug  36  ft.  X 
42  ft.  ? 

3.  At  60/  per  yd.,  what  will  be  the  least  cost  to  carpet 
a  room  14  ft.  X  16  ft.  with  ingrain  carpet,  using  only  full 
breadths,  and  no  waste  for  cutting  ? 

4.  Reduce  .875  of  a  bushel  to  lower  denominations. 

5.  How  many  bushels  will  a  bin  contain  that  is  9  ft. 
long,  4  ft.  wide,  and  6  ft.  deep  ? 

6.  How  much  will  a  piece  of  land  20  rd.  by  18  rd.  cost 
at  $116  per  A.  ? 

7.  Find  the  cost  of  a  Brussels  carpet  (27  in.  wide)  at 
$1.15  per  yd.  for  a  room  16  ft.  by  23  ft.,  breadths  to  run 
crosswise. 

8.  At  $.60  per  sq.  yard,  what  will  it  cost  to  plaster 
sides  and  ceiling  of  a  room  18  X  12  X  8  ft.  ? 

9.  Leaving  Lockport,  I  travel  until  my  watch  is  1  h.  20 
min.  slow.     Which  way,  and  how  far,  have  I  travelled  ? 

10.  My  cistern  is  8  ft.  by  4^  ft.  When  the  water  is  27 
in.  deep,  how  many  barrels  of  water  is  there  in  the  cistern  ? 


TEST   QUESTIONS.  251 

398.    1.    Define  and  illustrate  decimal  ;  multiple  ;  quotient. 

2.  If  I  burn  a  pint  of  kerosene  every  night,  what  will 
a  three  weeks'  supply  cost  me  at  15  cents  a  gallon  ? 

3.  Find  the  sum  of  J  mi.  ^  rd.  |  ft. 

4.  How  many  boards,  each  15  feet  long,  will  be  required 
to  build  56T4T  rods  of  fence  four  boards  high  ?     Analyze. 

5.  Find  the  value  of  |  of  a  chest  of  tea  weighing  57  J 
pounds,  at  $lj  per  pound. 


6.  Solve  i  LX32X_96XJL  =  ? 

192  x  21  x  28  X  55  X  8 

7.  How  many  times  will  a  wheel  12  ft.  4  in.  in  circum- 
ference revolve  in  going  10  miles  ? 

8.  How  many  days  must  a  laborer  work,  at  $1.12£  a 
day,  to  pay  for  6  cords  of  wood,  at  $3.37±  per  cord  ? 

9.  A  man  was  born  Feb.  29,  1844,  and  died  Mar.  15,  1880. 
How  many  birthdays  did  he  have  ?     What  was  his  age  ? 

10.  What  is  the  product  of  12  millionths  multiplied  by 
12  thousandths  ? 

399.  1.  How  many  pickets  3  in.  wide,  placed  3  in.  apart, 
will  be  required  for  a  fence  around  a  rectangular  yard  4  rd. 
6  ft.  long,  and  3  rd.  8  ft.  wide  ? 

2.  A  farmer  has  a  piece  of  land  containing  7}£  acres, 
fenced  in  the  form  of  a  rectangle,  its  length  being  twice 
its  width.     What  are  the  dimensions  of  rectangle  ? 

3.  Oswego  County  has  an  area  of  970  square  miles,  and 
a  population  of  71780.     What  is  the  population  to  the  square 
mile  ?     How  many  acres  could  be  given  to  each  one  of  the 
entire  population  ? 

4.  Oswego  is  in  longitude  76°  35'  W.,  Albany,  73°  32' 
W.     What  is  the  difference  in  their  longitude  ?     When  it 
is  noon  in  Albany,  what  o'clock  is  it  in  Oswego  ? 


252  SENIOR   ARITHMETIC. 

5.  What  will  be  the  cost  of  carpeting  a  room  18  ft.  long 
and  12  ft.  wide  with  Brussels  carpet  £  yd.  wide,  at  So/  a 
yd.,  the  strips  to  run  lengthwise  of  the  room,  and  allowing 
4  in.  to  be  turned  under  ? 

6.  At  $25  per  thousand,  what  is  the  value  of  16  planks, 
each  18  ft.  long,  6  in.  wide,  2^  in.  thick  ? 

7.  Find  the  cost  of  5  pieces  of  timber,  each  48  ft.  long, 
9  in.  by  12  in.,  at  $1.50  per  hundred  bd.  ft. 

8.  How  many  board  feet  of  lumber  will  be  required  to 
fence  a  lot  80  ft.  by  40,  the  boards  being  10  ft.  by  6  in., 
and  the  fence  4  boards  high  ? 

9.  How  many  board  feet  will  it  take  to  cover  the  top 
of  a  tank  14  ft.  long,  6  ft.  wide,  with  boards  2  in.  thick  ? 

10.  A  man  sold  two  bushels  of  strawberries  as  follows : 
to  Mrs.  A.  he  sold  T^  of  the  berries,  to  Mrs.  B.  f ,  and  the 
remainder  to  Mrs.  C.  How  many  quarts  did  Mrs.  C.  buy  ? 

400.  1.  Two  telegraph  stations  are  18  miles,  224  rods 
apart.  If  the  telegraph  poles  between  the  stations  are  8 
rods  apart,  how  many  poles  will  be  needed,  and  how  much 
will  they  cost  at  50/  apiece  ? 

2.  What  is  the  value  of  a  triangular  piece  of  land,  hav- 
ing a  base  of  60  chains  and  an  altitude  of  40  chains,  at  $60 
per  acre  ? 

3.  How  many  times  can  a  dish  holding  2  qt.  1  pt.  be 
filled  from  a  jar  holding  3  gal.  2  qt.  1  pt.  ?     How  much 
will  be  left  in  the  jar  ? 

4.  Find  cost  of  carpeting  a  room  24  ft.  long  and  18  ft. 
wide,  with  carpet  27  inches  wide,  the  strips  running  length- 
wise of  the  room,  cost  of  carpet  being  $1.65  a  yard,  and  no 
loss  in  matching  the  figures. 

5.  After  spending  $46|,  I  had  f  of  my  money  left. 
How  much  had  I  at  first? 


TEST   QUESTIONS.  253 

6.  A  man  traded  7  wagons  at  $77  1-  each  for  84  bbl.  of 
flour  ;  what  was  the  flour  per  barrel  ? 

7.  What  is  the  capacity  in  liters  of  a  cistern  1.5  meters 
long,  9  decimeters  wide,  and  86  centimeters  deep  ? 

8.  How  many  bricks  8  in.  long,  4  in.  wide,  and  2  in. 
thick  will  it  take  to  pave  a  section  of  street  200  ft.  long, 
36  ft.  wide,  the  bricks  being  placed  on  their  edges  ? 

Ho\v  much  will  the  bricks  cost  at  $7.35  per  M.? 

9.  What  is  the  depth  of  a  cubical  bin  which  contains 
300  bu.  of  wheat  ? 

10.    The  distance  around  a  circular  park  is  1^  miles. 
How  many  acres  does  it  contain  ? 


401.  1.  How  many  blocks  }£  of  a  ^°°^  l°n£  can  ^e  cu^ 
from  a  board  22  ft.  long  ? 

2.  How  many  poor  families  can  be  supplied  with  f  of 
a  ton  of  coal  each  from  12  tons  ? 

3.  How  many  pairs  of  tray-cloths,  each  containing  f  of 
a  yard,  can  be  cut  from  15  yards  of  linen  ? 

4.  In  how  many  months,  paying  $J  per  week,  will  a 
debt  of  $36  be  paid  ? 

5.  }  is  what  part  of  |  ? 

6.  A  37-gallon  cask  is  g  full;  6^  gallons  being  drawn 
off,  how  full  will  it  be  ? 

7.  If  from  a  piece  of  cloth  containing  96  yd.  you  sell 
24  1  yd.,  what  fractional  part  of  the  piece  remains  ? 

8.  11  1  bushels  are  what  fraction  of  15|  bushels  ? 

9.  ^  is  what  part  of  §  of  |  ? 

10.  A  man  had  700  head  of  cattle.  He  sold  at  one  time 
50  head,  at  another  75  head.  What  fraction  of  the  whole 
did  he  sell  ? 


254  SENIOR   ARITHMETIC. 

402.  l.    How  many  cubic  feet  of  stone  will  it  take  to 
build  the  walls  of  a  cellar  36  ft.  long,  24  ft.  wide,  and  8 
ft.  high,  outside  measurement,  the  walls  being  18  in.  thick  ? 
How  much  will  the  stone  cost  at  $4.50  per  cord  ? 

2.  Find  the  diameter  of  a  wheel  whose  circumference  is 
50  feet. 

3.  If  1  bu.  3  pk.  6  qt.  of  walnuts  cost  $3.10,  what  is  the 
price  per  quart  ? 

4.  What  will  be  the  cost  of  5  gal.  3  qt.  1^  pt.  of  maple 
sirup  at  75  cents  per  gallon  ? 

5.  Find  the  cost  of  5362  pounds  of  coal  at  $4.50  per 
ton. 

6.  How  long  a  time  has  elapsed  since  the  first  message 
was  sent  by  telegraph,  May  29,  1844  ? 

7.  How  much  profit  will  there  be  in  buying  4  bu.  1  pk. 
6  qt.  of  cranberries  at  $2  a   bushel,  and  selling  them  at 
10  cents  a  quart  ? 

8.  How  many  days  will  a  6-ounce  bottle  of  medicine 
last  a  patient  who  takes  a  teaspoonful  three  times  a  day, 
a  teaspoon  holding  60  drops  or  minims  ? 

9.  Multiply  9  mi.  25  rd.  3  yd.  2  ft.  by  f. 

10.    Divide  110  mi.  149  rd.  3  yd.  2  ft.  6  in.  by  J. 

403.  1.    Multiply  25  yards  2  ft.  11  in.  by  16. 

2.  From  6  bu.  6  qt.  take  3  pk.  1  qt.  1  pt. 

3.  What  is  the  difference  in  time   between  June   16, 
1890,  and  Feb.  4,  1895? 

4.  What  will  it  cost  to  build  the  walls  of  a  cellar  that 
is  26  ft.  long  and  16  ft.  wide,  6£  ft.  deep,  the  wall  being 
18  in.  thick,  at  $1.50  a  perch  ? 


TEST   QUESTIONS.  255 

5.  A  field  is  16  ch.  10  links  long  and  5  cli.  wide.     How 
maiiy  acres  does  it  contain  ? 

6.  How  many  board  feet  in  24  joists,  10  in.  by  2  in.  by 
16  ft.,  and  what  are  they  worth  at  $11  per  M.  ? 

7.  What  is  a  pile  of  four-foot  wood  worth  that  is  16  ft. 
long  and  6  ft.  high,  at  $4.50  a  cord  ? 

8.  How  many  grains  in  5  Ib.  of  butter  ? 

9.  Reduce  12  cwt.  80  Ib.  6  oz.  to  the  decimal  of  a  ton. 
10.    Find  the  sum  of  184f,  372|,  19|. 

PERCENTAGE. 

404.    1.    Express  as   %  the  following:  .28;  .065;  3.07; 
.004. 

2.   Express  decimally  the  following:  \%\  6|%;  8%  ; 


3.  From  a  farm  of  144  acres  18  acres  were  sold.    What 
per  cent  of  the  farm  was  sold  ? 

4.  A  grocer  sold  eggs  at  12£  cents  a  dozen  and  gained 
25%.     What  was  the  cost  ? 

5.  A  man's  farm  cost  him  $5,400  ;  his  crop  of  potatoes 
yielded  him  in  cash  8%  of  the  cost  of  the  farm.     What 
was  the  value  of  his  potatoes  ? 

6.  If  a  merchant  pays  $.80  a  yard  for  a  roll  of  carpet, 
and  because  it  became  damaged  sells  it  for  $.65  a  yard, 
what  per  cent  does  he  lose  ? 

7.  Sent  my  agent  in  St.  Louis  $3017.60,  with  which  be 
is  to  purchase  flour  at  $4.00  per  bbl.,  after  deducting  his 
commission  at  2^   per  cent.     How  many  barrels   should 
I  receive  ? 

8.  If,  by  selling  36840  ft.  of  lumber  at  $21.12  per  M., 
you  gain  28  per  cent,  what  would  be  your  gain  or  loss  by 
selling  it  at  $17  per  M.  ? 


256  SENIOR   ARITHMETIC. 

9.  If  a  merchant  has  marked  an  article  for  sale  at  50 
per  cent  above  cost,  what  per  cent  will  he  deduct  from  the 
asking  price  if  he  sells  the  article  at  cost  ? 

10.  $7884.00  is  to  be  raised  by  taxation  in  a  certain 
school  district  The  taxable  property  of  the  district  is 
$584,000.  Find  the  rate  of  tax,  and  A's  tax,  whose 
property  is  assessed  at  $3850. 

405.  1.    From  \  of  a  week  take  \  of  a  day. 

2.  Eeduce  — 5  to  a  simple  fraction. 

12| 

3.  Define  base  and  rate. 

4.  How  many  hundredths  of  anything  is  1  of  it  ?  ^  of 
it  ?  i  of  it  ?  T^  of  it  ? 

5.  What  is  12%  of  1682? 

6.  Express  as  common  fractions  in  their  lowest  terms  • 
25%,  62i%,  124%,  16|%. 

7.  A  speculator  bought   2160   barrels   of   apples,  and 
upon  opening  them  found   15%    of   them   spoiled.     How 
many  barrels  did  he  lose  ? 

8.  A   farmer  sold   50   sheep,   which  was  25%   of   his 
whole  flock.     How  many  sheep  had  he  at  first  ? 

9.  My  income  this  year  is  $4028,  which  is  24%  less 
than  it  was  last  year.     How  much  was  it  last  year  ? 

10.  A  commission  merchant  sells  goods  to  the  amount 
of  $6895.  What  is  his  commission  at  3%  ? 

406.  1.    I  bought  two  houses  at  $3500  each,  and  sold 
one  at  a  gain  of  22%,  and  the  other  at  a  loss  of  22%. 
Bid  I  gain  or  lose  on  both  ?  and  how  much  ? 

2.  If  I  sell  for  $16  what  cost  $20,  what  per  cent  do  I 
lose? 


TEST   QUESTIONS.  257 

3.  If  I   buy  a  piano  for  $450,   and  sell  it  for  $600, 
what  per  cent  do  I  gain  ? 

4.  Define  insurance  ;  premium  ;  taxes. 

•5.    What  will  be  the  cost  of   insuring  a  quantity  of 
wheat  valued  at  $8,450,  at  f  %  ? 

6.  The  preniium  for  insuring  a  schoolhouse,  at  the  rate 
of  1|%,  was  $75.     For  what  sum  was  it  insured  ? 

7.  The   town   of   B   is  to  be   taxed    $3,700   to   build 
a  bridge;  the  taxable   property  is  valued   at  $l,£f50,000. 
What  will  be  the  rate  of  taxation,  and  the  tax  of  Mr.  A., 
whose  property  is  valued  at  $5,000  ? 

8.  What  is  the  duty,  at  25%,  on  4796  pounds  of  Russia 
iron,  worth  10  cents  a  pound  ? 

9.  What  number  increased  by  25%  of  itself  is  506.25  ? 
10.    Find  the  net  cost  of  a  bill  of  goods  amounting  to 

$3,750  at  10%  discount,  and  4%  off  for  cash. 

407.    l.    An  agent  sold  4,250  yd.  of  calico  at  3|/  per 
yard.     What  was  his  commission  at  2|  %  ? 

2.  A  real  estate  broker,  who  charges  4%  commission, 
receives  $224  for  selling  a  house.     What  price  is  paid  for 
the  house  ? 

3.  If  $8,240  is  sent  to  an  agent  to  cover  the  amount 
of  his  purchase  and  his  commission  of  3%,  what  is  the 
amount  of  his  purchase  ? 

4.  A  hotel  is  insured  for  $90,000  at  2£%  for  3  years. 
What  is  the  annual  cost  of  insurance  ? 

5.  A   man's  weight   is  180   pounds,    and   he   is   20% 
heavier  than  his  brother.     What  is  his  brother's  weight  ? 

6.  A  bill  for  hardware  amounting  in  gross  to  $2,537.75 
is  subject  to  discounts  of  40%,  10%,  and  5%.     What  is 
the  net  amount  ? 


258  SENIOR   ARITHMETIC. 

• 

7.  If  you  remove  the  decimal  point  from  the  number 
6.45,  what  effect  does  it  produce  upon  the  number  ? 

8.  If  from  the  same  number  you  take  the  period  from 
after  the  6  and  place  it  before  the  6,  what  will  be  the 
effect  ? 

9.  At  $12.75  a  ton  what  will  3265  pounds  of  hay  cost  ? 
10.   A  tree  measures  8.2  ft.  in  circumference.     What  is 

the  diameter  ? 


408.  1.  Find  \%  of  $12.00;  ^%  of  2000  bushels  of 
corn;  200%  of  5  dozen  eggs;  f  of  1  per  cent  of  100 
tons  of  coal. 

2.  What  fraction  increased  by  25  per  cent  of   itself 
equals  \\  ? 

3.  What  is  the  effect  upon  the  quotient  when  both  the 
dividend  and  the  divisor  are  multiplied  by  the  same  num- 
ber ? 

4.  Express   as   fractions  in  lowest  terms,  8^%, 


5.  Express  as  per  cent,  using  the  sign,  .1352,  },  2,  5^0. 

6.  Express  as  decimals,  ,JT,  J,  \%,  20%,  15J%. 

7.  What  per  cent  of  the  number  of  days  in  February, 
1896,  is  the  number  of  days  in  January,  1896  ? 

8.  My  house  cost  $6000,  which  was  400  per  cent  more 
than  I  paid  for  the  lot.     Find  the  cost  of  both. 

9.  After  spending  $14  for  a  suit  of  clothes,  a  man  had 
$126  left.     What  per  cent  of  his  money  did  he  spend  ? 

10.  An  agent  purchased  8^  tons  of  sugar  at  3^  cents 
per  pound  on  3%  commission.  Find  the  cost  of  the  sugar, 
including  commission. 

409.  1.  What  is  the  rate  of  taxation  on  $1000  when 
$147000  is  raised  on  $35,000,000  ? 


TEST   QUESTIONS.  259 

2.  A  man  selling  cloth  at  $4.20  per  yard,  gained  20%. 
Had  he  sold  it  at  $3.60  per  yard,  would  he  have  gained  or 
lost  ?  and  what  per  cent  ? 

3.  If  f  of  a  mill  is  worth  $10000,  what  is  1  of  the 
remainder  worth  ? 

4.  Bought  a  horse  for  $160J,  and  sold  it  for  £  of  its 
cost.     How  much  did  I  lose  ? 

5.  Define  least  common  multiple;   improper  fraction; 
prime  factor. 

6.  Simplify  ?| . 

7.  Find  the  cost  of  10  sticks  of  timber,  each  16  feet 
long,  14  inches  wide,  and  10  inches  thick,  at  $16.50  per  M., 
board  measure. 

8.  How  many  gallons  will  a  cistern  hold  that  is  12  ft. 
long,  8  ft.  wide,  and  6  ft.  deep  ? 

9.  If  9|  yards  of  cloth  are  worth  $24.375,  what  is  the 
value  of  16 1  yards  at  the  same  rate? 

10.    Name  the  unit  of  weight  in  the  metric  system,  and 
give  the  table  in  which  that  unit  occurs. 

410.    1.    I  spend  65  per  cent  of  my  salary,  but  am  able 
to  save  $980  ;  how  much  do  I  spend  ? 

2.  How  much  must  I  send  my  agent  that  he  may  buy, 
at  1^  per  cent  commission,  400  bbl.  flour  at  $6.75  per  bbl.? 

3.  Given  the  amount  and  percentage,  write  the  formula 
for  rinding  each  of  the  other  terms. 

4.  What  are  like  numbers  ?     Unlike  numbers  ? 

5.  Write  an  abstract  number.     Give  definition  of  ab- 
stract number. 

6.  Write  in  words,  2300406^000960. 

7.  What  kind  of  number  is  4.6  bushels  ? 


260  SENIOR    ARITHMETIC. 

8.  A  father  divided  his  property  as  follows  :  to  his  son 
John  he  gave  1,  to  his  daughter  Susan  ^,  to  his  wife  1,  and 
the  rest,  which  was  $13000,  to  endow  a  school.    What  was 
the  value  of  his  estate  ? 

9.  I  own  a  house  that  cost  me  $3000.     It  cost  me  to 
insure   it  for  3  years  $24.     The  average  yearly  cost  of 
repairs  is  $50.     The  average  yearly  tax  is  2%  of  the  cost. 
I  can  get  5%   per  annum  for  the  $3000  invested.     The 
house  will  last  60  years.     I  receive  in  rent  for  the  house 
$300  per  annum.     If  these  conditions  are  constant,  how 
much  will  I  gain  or  lose  in  60  years  ? 

10.    A  father  is  39  years  old  and  his  daughter  13  ;  what 
per  cent  of  the  father's  age  is  the  daughter's  ? 


411.    1.   Write  these  per  cents  as  hundredths*   2 
20%,  12i%. 

2.  How  many  per  cent  of  a  number  is  0.20  ?  0.75  ? 
.121  ?  1.40  ? 

3.  What  fractions  of  a  number  (in  lowest  terms)  are 
these  per  cents:  16§  %  ?  75%  ?  33i%  ?  100%  ?  and  175%  ? 

4.  Express  as  hundredths  and  as  common  fractions  : 


5.  From  a  stack  of  hay  7  T.  11  cwt.  were  sold,  which 
was  75i%  of  the  whole.     How  much  did  the  stack  con- 
tain before  the  sale  ? 

6.  A  lawyer  collected  65%  of  a  debt  of  $1260,  and 
charged  5%  commission  on  the  sum  collected.     What  did 
the  creditor  receive  ? 

7.  If  a  hat  that  cost  $5  be  sold  for  $9,  what  is  the 
gain  per  cent  ? 

8.  How  many  days   from   Sept.  16,  1892,  to  Feb.  12, 
1894? 


TEST   QUESTIONS.  261 

9.    874  is  33^%  less  than  what  number? 
10.    Required  the  cu.  feet  of  a  box  6  ft.  6  in.  by  4  ft.  9  in. 
by  3  ft.  3  in. 

412.  1.   Write  the  following  numbers  and  add:  six  thou- 
sand sixteen  and  sixty-five  thousandths,  four  hundred  one 
thousand  forty-one  and  one-tenth,  six  hundred  one  and  nine 
hundredths,  ten  thousand  one  hundred  seventeen  and  nine 
hundred  three  thousandths,  forty-nine  hundred  forty-nine 
and  nine-tenths. 

2.  Write  in  words  83.493, 7007T7o,  1001001.01,  90019T^. 

3.  Find  the  number  of  which  160  is  f . 

4.  Find  the  exact  number  of  days  from  July  4, 1893,  to 
to-day. 

5.  Multiply  7  Ib.  8  oz.  15  pwt.  by  15. 

18  X  963  x  44  x  27  X  2800  _ 
63  X  88  X  105  X  1926  X  45 

7.  Define  commission,  also  brokerage ;  and  state  on  what 
sum,  or  value,  both  are  computed. 

8.  Express  decimally  27f  and  ^\.     Find  their  product 
as   decimals,  and  as   common   fractions,   expressing   both 
answers  decimally. 

9.  Fruit  was  sold  at  12^/  per  quart,  which  was  200  per 
cent  of  its  cost.     What  was  the  cost  per  bushel  ?  and  what 
was  the  rate  per  cent  of  profit  ? 

10.    An  agent  sold  840  bu.  grain  at  60/  per  bushel.     His 
commission  was  $15.12.     Find  the  rate  of  commission. 

413.  1.    A  man  owes  you  a  debt  of  $2160,  which  he  de- 
clines to  pay.     Your  lawyer  succeeds  in  collecting  70  per 
cent  of  the  debt,  and  charges  5  per  cent  commission  for 
his  services.     What  sum  do  you  receive  ? 


262  SENIOR    ARITHMETIC. 

2.  A  manufacturer  sent  $1295.27  to  a  commission  mer- 
chant who  charges  3  per  cent  commission,  instructing  him 
to  purchase  wool  at  $0.33^  per  pound.     How  many  pounds 
of  wool  will  be  received  ? 

3.  A  farm  was  sold  for  $8000,.  which  was  20  per  cent 
less  than  its  real  value.     If  it  had  sold  at  $12000,  what 
per  cent  above  its  real  value  would  it  have  brought  ? 

4.  A  commission  merchant  sold  for  a  farmer  6000  Ib. 
pork  at  8J/  per  pound.      He  charged  \\%  commission  for 
selling,  and  paid  $18.80  for  freight.     How  many  feet  of 
pine  boards  at  $25  per  1000  ft.  could  he  purchase  with  the 
proceeds  of  the  pork,  after  deducting  1  per  cent  commis- 
sion for  buying  ? 

5.  Reduce  to  simple  fraction  in  lowest  terms : 

T2T    Of    12* 

I  xf +  §' 

6.  What  per  cent  of  3  is  f  ?     Of  f  is  f  ?     Of  80  is  50  ? 

7.  A  drover  sold  250  sheep  for  $1150,  which  was  15% 
more  than  they  cost.     What  was  the  cost  per  head  of  the 
sheep  ? 

8.  If  20%  be  lost  on  a  ton  of  rye  straw  sold  for  $19.20, 
what  is  the  cost  of  the  straw  per  ton  ? 

9.  How  many  per  cent  of  a  number  is  0.15  ?  0.06|  ? 
0.50  ?     2.25  ? 

10.  What  common  fraction  of  a  number  in  its  lowest 
terms  is  20%  ?  50%  ?  6J%  ?  66§%  ?  160%  ? 

414.  1.  A  man  sold  $8400  worth  of  merchandise,  and 
had  30%  of  his  stock  left.  What  was  his  entire  stock 
worth  ? 

2.  A  merchant  sold  goods  at  20%  and  5%  off,  and  still 
made  20%  on  the  cost.  What  was  the  cost  price  of  a  book 
that  was  marked  $1.00  ? 


TEST   QUESTIONS.  263 

3.  Bought  1000  pounds  of  butter  at  18^,  and  sent  it  to 
an  agent  who  sold  it  at  21/  on  a  5%  commission.     What 
was  my  rate  of  gain  ? 

4.  Mr.  Brown  has  a  flock  of  940  sheep  in  three  fields. 
In  the  first  are  20%  of  the  entire  flock,  in  the  second  40%, 
and  the  remainder  in  the  third.      How  many  sheep  are 
there  in  each  field  ? 

5.  A  lady  has  a  salary  of  $825  a  year ;  she  spends  20% 
of  it  for  board,  35%  of  it  for  other  expenses,  and  saves  the 
remainder.     What  sum  does  she  save  ? 

6.  What  per  cent  of  a  leap  year  is  the  time  from  Wash- 
ington's Birthday  to  the  Fourth  of  July  ? 

7.  The  Barber    Asphalt  Company  engaged  to   pave  a 
street  5  miles  long  at  $55000  a  mile.     If  the  actual  cost 
be  $130  per  rod,  what  is  the  gain  per  cent  ? 

8.  A  commission    merchant    charges  1|%   for  selling, 
and  2|%  for  guaranteeing  the  payment  of  the  money.    His 
commission  on  a  certain  transaction  amounted  to  $384.75. 
Required  the  amount  of  the  sale. 

9.  I  bought  1100  tons  of  coal  at  $3^  per  ton.     I  sold 
40%  of  it  at  a  gain  of  50%,  40%  of  the  remainder  at  a 
gain  of  35%,  and  lost  10%   on  the  rest.     Wliat  was  my 
actual  gain  ? 

10.    An  article  bought  at  18%  below  the  asking  price  is 
sold  for  the  asking  price.    What  is  the  gain  per  cent  ? 

INTEREST   AND    DISCOUNT. 

415.    1.    Find  the  amount  of  $875  for  1  year,  4  months, 
and  12  days,  at  6  per  cent  interest. 

2.  Find  the  interest  on  $128.45  from  March  2,  1895, 
to  Dec.  14,  1895,  at  6  per  cent. 

3.  A  pile  of  wood  256  feet  long,  4  feet  wide,  and  5  feet 
high  is  sold  for  $160.     What  is  the  price  per  cord  ? 


264  SENIOR   ARITHMETIC, 

4.  Define  per  cent ;  interest ;  proper  fraction. 

5.  State  the  difference  between  a  prime  and  a  composite 
number. 

6.  Find  the  cost  of  6  gal.  3  qt.  and.  1  pt.  of  sirup  at 
46  cents  per  gallon. 

7.  1521  is  how  many  times  13  ? 

8.  What  is  the  interest  on  $1200  for  2  yr.  3  mo.  18 
da.  at  6%  ?     The  amount  ? 

9.  What  is  the  interest  on  $1240  from  March  3  to  Aug. 
28,  at  6%  ? 

10.  Write  the  United  States  rule  for  computing  the 
amount  due  on  a  note  when  partial  payments  have  been 
made. 

416.  1.  In  what  time  will  $3960  earn  $770  at  5%, 
simple  interest  ? 

2.  If  $675,  at  simple  interest,  gain  $172.80  in  3  years, 
2  months,  12  days,  what  is  the  rate  of  interest  ? 

3.  When  interest,  time,  and  rate  are  given,  how  may  the 
principal  be  found  ? 

4.  Define  true   present  worth  and   true  discount  of  a 
debt.    Define  compound  interest,  and  make  and  solve  an 
example  to  illustrate  your  definition. 

5.  A  merchant  sells  goods  amounting  to  $6784.00  on  a 
year's  credit.     If  money  is  worth  8%,  what  sum  should  he 
accept  in  payment  of  the  bill  6  months  before  it  becomes 
due? 

6.  Write  a  negotiable  promissory  signed  by  James  Fox 
for  $875.60  due  90  days  from  date,  payable  to  yourself,  at 
a  bank.    Name  (a)  the  payee  ;  (b)  the  drawer  ;  (c)  the  date 
when  the  note  matures  (becomes  due).    What  words  on  the 
note  make  it  negotiable  ?     What  does  negotiable  mean  ? 


TEST    QUESTION'S"".----*—.  265 

7.  If  you  should   sell  the   note  (Ex.  6)  to   Mr.   F.  P. 
Weaver,    what    indorsement    must    you   write    upon    it  ? 
Where  should  indorsements  be  written  ? 

8.  If  the  note  is  not   paid  until  Sept.  15,  1895,  how 
much  interest  will  then  be  due  on  it  ? 

9.  A  farmer  expended  $5580  in  improvements  on  his 
farm,  which  was  24%  more  than  J  of  the  cost  of  the  farm. 
Find  the  cost  of  the  farm. 

10.  Principal,  interest,  and  time  being  given,  how  is  the 
rate  found  ? 

417.  1.  Find  the  amount  of  $496.85  for  2  years,  4 
months,  and  15  days  at  4  per  cent. 

2.  How  long1  will  it  take  $750  at  6  per  cent  to  gain 
$67.50  interest  ? 

3.  A  dealer  bought  65  lawn-mowers  at  $4.25  each,  and 
sold  them  at  $3.87^  each.     What  per  cent  did  he  lose  ? 

4.  If  a  cellar  is  38  ft.  long  and  28  ft.  wide  inside  the 
wall,  and  the  wall  is  8  ft.  high  and  18  in.  thick,  how  many 
cubic  yards  of  masonry  does  the  wall  contain  ? 

5.  What  per  cent  of  a  number  equals  f  of  the  number  ? 
What  part  of  a  number  equals  33^  per  cent  of  it  ? 

6.  Write  decimally,  6%  ;   one  hundred  six  per  cent. 

7.  A  town  6  miles  long  and  4J  miles  wide  is  equal  to 
how  many  farms  of  80  acres  each  ? 

8.  What  number  must  be  subtracted  from  four  hundred 
sixty-seven  thousand  six  hundred  thirty-three  to  make  it 
exactly  divisible  by  758  ? 

9.  Find  the  amount  of  $535.20  for  2  yr.  4  mo.  18  da. 
at  5  per  cent,  simple  interest. 

10.  Give  formula  or  rule  for  finding  the  base  when  rate 
per  cent  and  difference  are  given.  Form  and  write  such  a 
problem. 


266  SENIOR    ARITHMETIC. 

418.  1.    Find  the  interest  of  8263.75  for  1  yr.  3  mo.  16 
da.  at  5%. 

2.  Make  a  30-day  bank  note  dated  Jan.  20,  1896,  for 
$600,  payable  at  some  bank.     Find  date  of  maturity,  the 
discount,  and  proceeds  if  discounted  on  the  date   of   the 
note.     (Make  the  note  on  a  separate  piece  of  paper,  and 
have  it  properly  indorsed.) 

3.  What  is  the  present  worth  of  a  note  due  in  1  yr.  6 
mo.  ? 

4.  In  what  time  will  $600  gain  $30  interest  at  6%  ? 

5.  What  will  $300  amount  to  in  4  years  compounded 
annually  at  4%  ? 

6.  An  agent  says  he  will  insure  your  house  for  3  years 
at  65.     What  does  he  mean  by  "  at  65  "  ? 

7.  Define  interest ;  principal ;  usury ;  compound  interest. 

8.  Find  the  amount  of  $684.50  for  3  yr.  4  mo.  at  7%. 

9.  Compute  the  interest  of  $1250  for  2  yr.  5  mo.  18  da. 
by  the  six  per  cent  method. 

10.    What  is  the  interest  on   a  note  for  $515.62,  dated 
March  1,  1885,  and  payable  July  16,  1888  ? 

419.  1.    A  note  for  $710.50,  with  interest  after  3  mo. 
at  8%,  was  given  Jan.  1,  1884,  and  paid  Aug.  13,  1886. 
What  was  the  amount  due  ? 

2.  What  sum  of  money  will  gain  $173.97  in  4  yr.  4  mo. 
at  6%  ? 

3.  What  is  the  legal  rate  of  interest  in  this  State? 

4.  Find  the  exact  interest  of  $950  at  5%  for  98  days. 

5.  What  principal  will  amount  to  $1531.50  in  1  yr.  3 
mo.  6  da.  at  6%  ? 

6.  At  what  rate  will  $1500  amount  to  $1684.50  in  2 
years,  18  days  ? 


TEST   QUESTIONS.  267 

7.  In  what  time  will  $840  gain  $78.12  at  6%  ? 

8.  Ho\v  long  will  it  take  any  sum  of  money  to  double 
itself  at  4%  ? 

9.  Find  the  compound  interest  of  $460  for  1  yr.  5  mo. 
24  da,  at  6%  interest,  payable  semi-annually. 

10.  If  f  of  an  acre  of  land  costs  $15,  what  will  10 £  acres 
cost  ? 

420.  1.  Name  four  different  forms  of  reduction  of  com- 
mon fractions.  Illustrate  one  of  them  to  show  that  the 
value  of  the  fraction  remains  unchanged. 

2.  Define  simple  interest,  true  discount,  and  bank  dis- 
count.   How  does  bank  discount  differ  from  interest?    How 
does  it  differ  from  true  discount  ? 

3.  Define  cancellation,  and  state  the  principle  of  arith- 
metic that  authorizes  its  use. 

4.  Find  the  amount  of  $575.87^  at  5  per  cent,  simple 
interest,  from  Aug.  5,  1883,  to  March  17,  1885. 

5.  What  principal  will  earn  $71.68  in  2  years,  4  months, 
at  6  per  cent,  simple  interest. 

6.  At  what  rate,  simple  interest,  will  $175  amount  to 
$203.35  in  3  yr.  7  mo.  6  days  ? 

7.  In   what   time   will   $4260  earn  $873.30,  at  6  per 
cent  ? 

8.  A  60-day  note  for  $610.25,  dated  June  12,  1889, 
was  discounted  in  bank,  July  1,  at  6  per  cent.     Find  the 
term  of  discount,  discount,  and  proceeds. 

9.  Having  purchased  a  horse  for  $125,  you  wish  to 
borrow  that  amount  at  bank  for  6  mo.     Write  your  own 
note,  indorsed   by   your  parent  as   security,  for  the    sum 
which,  discounted  to-day,  will  give  $125  as  proceeds  of  the 
note. 


268  SENIOR    ARITHMETIC. 

10.  A  stock  of  goods  was  owned  by  three  parties.  A 
owned  f,  B  f ,  and  C  the  remainder.  The  goods  were  sold 
at  a  profit  of  $4260.  What  was  each  one's  share  of  the  gain  ? 

421.  1.  A  horse  is  offered  me  for  $350  cash,  or  for 
$382.50  to  be  paid  in  4  mo.  What  can  I  save  by  paying 
cash,  the  rate  of  interest  being  6%  ? 

2.  Which  is  the  more  profitable,  and  how  much,  money 
being  worth  5%,  to  buy  a  house  for  $5940  on  2  years' 
credit,  or  for  $5219.30  on  6  months'  credit? 

3.  A  note  dated  June  20,  1893,  and  bearing  interest  at 
6  per  cent,  was  paid  Aug.  15,  1895.     The  face  of  the  note 
being  $68.45,  what  was  the  amount  paid  ? 

4.  Bought  150  front  feet  of  land  at  $40  per  front  foot, 
paid  $116  city  taxes,  $32  county  taxes,  and   $320  local 
taxes ;  at  the  end  of  two  years  I  sold  for  $60  per  front 
foot.      Keckoning  interest  at  6%  on   the  purchase  price, 
did  I  gain  or  lose  by  the  transaction  ?  and  how  much  ? 

5.  A  man  wishes  to  pay  me  $3252.56.     Not  having  the 
money,  he  borrows  it  from  a  bank  by  giving  his  note  for  48 
days  at  4%.     For  what  sum  does  he  draw  the  note?     No 
grace. 

7. 

Buffalo,  N.  Y.,  Q/Al.  3,  189^. 

after  date,  &  promise  to  pay 

order, 
Value 

0  (f  fl  IOO 

received. 

This  note  was  discounted  May  4,  1896.     Find  the  proceeds. 


TEST   QUESTIONS.  269 

7.  Required     the    simple     interest    and     amount     of 
$7231.289  for  3  yr.  8  mo.  15  days  at  8%. 

8.  Face  of  note  $750.     Time  60  da.     Rate  6%.     To 
find  proceeds. 

9.  Write  the  following  in  a  note  properly,  and  find  the 
maturity  and  proceeds  :  Face,  $600 ;  date,  -April  3,  1896 ; 
due  in  90  days ;  discounted  at  bank,  May  20,  1896,  at  6%, 

with  grace. 

10. 

Saratoga  Springs,  N.  Y.,  (Q/€<f  3,  i8g£. 
nd  after  date,  ©*  promise  to  pay  to  the 

&/rtne  Wt&Mdtintt   Dollars,  at  the  First  National  Bank. 
Value  received. 

Find  proceeds,  if  discounted  at  6%,  Dec.  3,  1895. 

STOCKS   AND   AVERAGE   PAYMENTS. 

422.    1.    How  many  shares  of  stock  at  80  can  I  buy  for 
$2550  ? 

2.  I  sold  two  houses  for  $2400  each.     On  one  I  gained 
10%,  on  the  other  I  lost  10%.     How  much  did  both  cost 
me  ?    Did  I  gain  or  lose  in  the  whole  trade  ?  and  how  much  ? 

3.  Find  the  cost  of  40  shares  of  American  Express  Co. 
stock  at  105^,  brokerage  1%. 

4.  A  mining  company  declares  a  dividend  of  8%   per 
annum  on  its  stock.     What  is  the  nominal  value  of  a  man's 
shares  who  gets  $864  as  his  semi-annual  dividend  ? 

5.  If  the  stock  of  a  railway  company  sells  at  5%  above 
par,  what  will  25  shares  cost  ? 

6.  If  I  invest  $21,008  in  5%  bonds  at  104,  what  will 
be  my  annual  income  ? 


270  SENIOR    ARITHMETIC. 

7.  Sugar  bought  at  5  cents  a  pound  was  sold  for  6£ 
cents ;  what  per  cent  was  gained  ? 

8.  What  sum  invested  in  4  per  cent  stock  will  yield  an 
annual  income  of  $320,  if  you  purchase  stock  at  par  ? 

9.  What  would  be  your  investment,   if    the  stock  is 
worth  15  per  cent  above  par  ? 

10.  A  man  invested  his  money  in  6  %  railroad  stocks,  and 
received  $300  semi-annually.  What  was  the  sum  invested  ? 

423.  1.  What  sum  must  be  invested  in  stocks  bearing 
6 1  per  cent  interest,  at  105  per  cent,  to  produce  an  annual 
income  of  $1000  ?  Solve  by  cancellation. 

2.  Define  brokerage,  certificate  of  stock,  par  value,  pre- 
mium  (as  used    in  stocks    and    investments).     What  are 
bonds  ?     Name  some  of  the  different  classes  of  bonds. 

3.  What  income  will  be  realized  from  investing  $4190.63 
in  5%  stock,  purchased  at  7%  discount,  if  I  pay  J%  for 
brokerage  ? 

4.  What  is  the  value  of  31  shares  of  $500  each,  sold  at 
a  premium  of  2T§^%  ? 

5.  Which  is  more  profitable,  to  buy  8%  bonds  at  25% 
premium,  or  6%  bonds  at  10%  discount  ? 

6.  A  owes  B  $3000,  due  as  follows:  June  15,  $1500; 
Sept.  10,  $400 ;  Nov.  1,  $500 ;  Dec.  15,  $600.     B  accepts 
in  settlement  Oct.  26  a  note  for  9  months,  bearing  interest 
at  6%  for  the  amount  of  the  debt,  with  6%  interest  due 
him  at  that  date.     Find  face  of  note. 

7.  On  Jan.  1,  1895,  a  merchant  gave  three  notes :  one 
for  $500,  payable  in  30  days ;  one  for  $400,  payable  in  60 
days ;  and  one  for  $600,  payable  in  90  days.     What  is  the 
average  term  of  credit,  and  what  the  equated  time  of  pay- 
ment ? 


TEST    QUESTIONS.  271 

8.  E.  ft.  Smith  owes  J.  D.  Wilson  $2500,  due  Oct.  12, 
1896.     If  Mr.  Smith  pays  $500  Aug.  10,  and  $1000  Sept. 
25,  when  should  the  balance  be  paid  ? 

9.  A 'speculator  bought  X.  Y.  C.  stock  at  98^,  and  sold  it 
at  97-|,  and  lost  $187.50.     How  many  shares  did  he  handle  ? 

10.  Had  he  retained  his  stock  until  a  quarterly  divi- 
dend was  declared,  his  dividend  would  have  been  $312.50. 
What  was  the  annual  rate  of  dividend  ? 

424.  1.    State  why  securities  fluctuate  in  value. 

2.  Name  a  corporation. 

3.  What  does  a  stockholder  hold  to  show  that  he  has 
stock  in  a  company  ? 

4.  On  what  does  the  income  from  his  stock  depend  ? 

5.  W7hy  does  a  corporation  issue  bonds  ? 

6.  Find  the  present  worth  and  true  discount  of  $300, 
due  in  10  months,  at  6%. 

7.  Find  the  bank  discount  and  proceeds  of  a  note  of 
$730,  due  in  3  months,  at  6%. 

8.  What  is  the  face  of  a  note  at  2  months  and  18  days, 
which  yields  $2961  when  discounted  at  a  New  York  bank  ? 

9.  A  person  owning  |  of  a  piece  of  property,  sold  20% 
of  his  share.     What  part  did  he  then  own  ? 

10.  At  what  price  should  4^%  bonds  be  bought  to  make 
the  income  from  investment  equivalent  to  that  from  3% 
bonds  at  par  ? 

PROPORTION   AND   PARTNERSHIP 

425.  l.    What  is  ratio  ? 

2.  Read  the  following  :  3  :  15.     What  does  it  equal  ? 

3.  What  is  each  of  the  numbers  in  the  above  expression 
called  ? 


272  SENIOR    ARITHMETIC. 

4.  What  is  a  proportion  ? 

5.  Is  the  following  expression  a  proportion  ?     Explain 
why.     9: 12::  16:  24. 

6.  24  :  (     )  =  56  :  7.      Find  the  omitted  term. 

7.  If  8  men  can  do  a  piece  of  work  in  10  days,  in  how 
many  days  can  12  men  do  it  ? 

8.  If  3  men  in  12  days  of  10  hours  each  can  build  a 
wall  100  feet  long,  14  feet  high,  and  3  ft.  thick,  how  long 
will  it  take  4  men  working  8  hours  a  day  to  build  a  wall 
200  feet  long,  16  feet  high,  and  4  feet  thick? 

9.  If  it  takes.  5  men  4  hr.  24  min.  to  manufacture  400 
boxes,  how  much  time  will  8  men  require  to  perform  the 
same  work. 

10.  If  I)  of  an  acre  of  land  cost  $15,  what  will  10£  acres 
cost? 

426.  1.  50  men  in  7  da.  at  12  hours  a  day  dig  a  cellar. 
How  many  men  will  be  requiied  to  dig  a  similar  cellar  in 
21 1  da.  of  8  hr.  each? 

2.  A  and  B  enter  into  partnership,  A  with  $1800  and 
B  with  $900.     After  8  mo.   B   adds  $300  to  his  capital. 
Divide  a  profit  of  $840  between  them  at  the  end  of  the 
year. 

3.  A  bankrupt   owes    A    $350,  B  $680.50,  C  $65,   D 
$500,  E  $980.50;  his  property  nets  $1648.64.     How  much 
does  each  creditor  receive  ?     How  much  does  he  pay  on  a 
dollar  ? 

4.  What  is  the  ratio  of  7  to  8  ?     Of  2i  to  3J  ?     Of  $9 
to  $6  ? 

5.  If  20  men  can  mow  a  field  in  6  days,  in  how  many 
days  will  30  men  mow  it  ? 

6.  If  5  horses  eat  8  bu.  14  qt.  of  oats  in  9  days,  at  the 
same  rate  how  long  will  66  bu.  30  qt.  last  17  horses  ? 


TEST   QUESTIONS.  273 

7.  A  and  B  hired  a  pasture  for  $40  for  the  season.     A 
put  in  9  cows  for  4  mo.,  and  B  put  in  8  cows  for  8  mo. 
Other  conditions  being  the  same,  what  should  each  pay  ? 

8.  In  what  time  will  $10,000  yield  $1200  interest  at 
8%.     Solve  by  proportion. 

9.  If  the  antecedent  is  §  of  T95  of  5%,  and  the  ratio  is  f 
of  If  of  } |,  what  is  the  consequent  ? 

10.    Required  the  ratio  of  6^  cu.  ft.  to  11  §  cu.  ft. 

427.  1.  A,  B,  and  C  entered  into  partnership.  A  put 
in  $600  for  8  mo.,  B  $800  for  7  mo.,  C  $1500  for  4  mo. 
They  gained  $820.  What  was  each  one's  share  of  the  gain  ? 

2.  A,  B,  and  C  found  a  gold-mine,  and  after  developing 
it  sold  it  for  $64000.  They  agreed  to  divide  the  money 
according  to  the  time  each  had  worked.  A  worked  37 
days,  B  46  days,  and  C  39  days ;  for  extra  services  B  is  to 
receive  $1800,  and  C  $1200  additional.  How  much  does 
each  receive  ? 

3.  Three  men,  A,  B,  and  C,  enter  into  partnership.     Out 
of  a  gain  of  $1200,  C  takes  $500  and  B  $400.      A's  in- 
vestment is  $4500.     Find  B's  and  C's  investment. 

4.  Divide  $450    among  three   people  in   the  ratio  of 
3,  4,  and  8. 

5.  Three  persons  bought  a  block  for  $21000,  of  which 
A    paid    $9000,    B    $8000,   and   C    the   remainder.      They 
rented  it  for  $1400  a  year.     What  was  each  man's  share 
of  the  rent  ? 

6.  Forster,  Stull,  and  Furlong  made  8000  pairs  of  bi- 
cycle pedals  in  1895,  which  they  sold  for  $1.60  per  pair. 
The  pedals  cost  them  $1.15  per  pair.     If  Mr.  Forster  put 
in  $1000  Jan.  1,  Mr.  Stull  $1200  April  1,  and  Mr.  Furlong 
$900  May  1,  what  would  be  each  one's  share  of  the  gain 
after  drawing  out  the  original  investment  ? 


274  SENIOR    ARITHMETIC. 

7.  Four    men    purchased    a    city    block    for    $36,000. 
The    first    contributed    $20,000,    the    second    $7,000,    the 
third  $4,000,  and  the  fourth  $5,000.     They  sold  the  land 
at  an  advance  of  50%  on  the  purchase  price.     How  much 
was  each  man's  share  of  the  gain  ? 

8.  A,  B,  and  C  form  a  partnership  in  which  A  is  to 
furnish  no  capital,  but  give  his  whole  time  to  the  business, 
and    have    i-    the    profits.       B    furnishes    $10,000,    and   C 
$15,000.     Their  net  profit  at  the  end  of  a  year  is  $8000. 
What  is  each  partner's  share  ? 

9.  A,  B,  and  C  gain  in  business  together  respectively 
$700,  $1000,  and  $1500.     What  was   the  investment  of 
each  if  their  joint  capital  was  $16,000  ? 

10.  Smith,  Brown,  and  Jones  gain  in  trade  $9400. 
Smith  furnished  $10,000  for  5  months,  Brown  $9000  for 
6  months,  Jones  $7000  for  1  year.  Apportion  the  gain. 

INVOLUTION  AND  EVOLUTION. 

428.  1.  Define  involution ;  evolution ;  a  square ;  cube 
root. 

2.  Find  the  square  of  6|  ;  of  2.35. 

3.  Find  the  third  power  of  123. 

4.  Find  the  square  root  of  the  fraction  f  §§£. 

5.  What  is  the  distance  around  a  square  field  which 
contains  40  acres  ? 

6.  A  man  has  640  acres  of  land.     How  much  more  will 
it  cost  to  enclose  it  with  a  fence  at  $4  a  rod,  in  a  rectangu- 
lar form  512  rods  long  and  200  rods  wide,  than  it  would  if 
in  the  form  of  a  square  ? 

7.  What  is  the  length  of  one  side  of  a  cube  which  con- 
tains 8120601  cubic  inches  ? 

8.  Find  the  entire  surface  of  a  cube  whose  volume  is  42 
cu.  ft.  1512  cu.  in. 


TEST    QUESTIONS.  275 

9.  The  edge  of  a  cube  is  42  inches.     Find  the  length  of 
the  edge  of  another  cube  4  times  as  large. 

10.  If  16  cords  of  wood  be  piled  in  the  form  of  a  cube, 
what  will  be  the  length  of  one  of  its  edges  ? 

429.  1.  What  are  the  length  and  breadth  of  a  rectangu- 
lar Held  which  contains  60  acres,  the  length  of  which  is 
three  times  its  breadth  ? 

2.  A  rectangular  farm  of  300  A.  is  1\  times  as  long  as 
it  is  wide.     How  many  miles  of  fence  will  enclose  it  ? 

3.  A  bird  is  15  feet  above  a  monument  80  ft.  high.     A 
boy  is  145  ft.  from  the  bird.     How  far  is  the  boy  from  the 
base  of  the  monument  ? 

4.  How  far  is  it  between  the  extreme  corners  of  a  box 
10  ft.  square  and  6  ft.  deep  ? 

5.  Find  how  many  acres  in  a  lot  in  the  form  of  a  right- 
angled  triangle  whose  hypothenuse  is  50  rd.  and  the  base 
40  rd. 

6.  Find  the  diagonal  of  a  square  piece  of  land  equal  in 
area  to  a  rectangular  piece  whose  dimensions  are  80  rd.  by 
20  rd. 

7.  Wishing  to  know  the  height  of  a  church  steeple,  I 
find  it  casts  a  shadow  165  ft. ;  I  also  find  that  a  10-ft.  pole 
when  placed  perpendicular  casts  a  shadow  12^  ft.     What 
is  the  height  of  the  steeple  ? 

8.  A  house  is  36  ft.  wide,  and  the  ridge  of  the  roof  is 
12  ft.  above  the  plates.     How  long  are  the  rafters  ? 

9.  A  steamer  goes  due  north  at  the  rate  of  12  miles 
an  hour,  and  another  goes  due  east  at  the  rate  of  15  miles  an 
hour.     How  far  apart  will  they  be  at  the  end  of  8  hours  ? 

10.  If  a  pineapple  5  in.  in  diameter  costs  20/,  what  should 
be  the  cost  of  a  pineapple  of  similar  shape  6  in.  in  diameter  ? 


276  SEN  I  Oil    ARITHMETIC. 

M  ISC  EL  LA  NEOUS. 

430.    1.    The  sum  of  two   numbers   is  2120,  and  their 
difference  938.     What  is  each  number  ? 

2.  J.  &  R.  Ross,  New  York,  bought  of  A.  L.  Covert  & 
Co.,  Philadelphia,  the  following  articles,  June  20,  1881  :  15 
Nichols's  Geography  at  $0.65 ;  12  Meiklejohn's  Literature 
at  $0.80 ;  25  Bowser's  Geometry  at  $0.75  ;  15  Hawthorne 
&  Lemmon's  Literature  at  $1.12  ;  10  Thomas's  U.  S.  His- 
tory at  $1.00. 

They  paid  $25  in  cash,  and  returned  books  to  the  amount 
of  $10.     Make  out  bill  showing  entire  statement. 

3.  Oswego,  N.Y.,  contains  22,000  inhabitants.     If  each 
inhabitant  should  contribute  one  cent  per  week  for  fifty- 
two  weeks  towards  the  erection  of  a  soldiers'  monument, 
how  expensive  a  monument  could  be  built  at  the  end  of 
the  year  ? 

4.  The  State  of  New  York  has  7746  miles  of  railroad, 
which  cost  $588,672,762.     Find  the  average  cost  per  mile. 

5.  The  sum  of  three  numbers  is  96  :  the  least  is  4^, 
and  greatest  37|.     Find  the  other  number  and  the  product 
of  the  three  numbers. 

6.  $9,000,000  has  recently  been  appropriated  for  im- 
proving the  Erie  Canal.     If  it  is  352  miles  long,  how  many 
dollars  may  be  expended  on  each  mile  ? 

7.  Find  the  least  common  multiple  of  24,  60,  75,  120. 

8.  What  is  the  smallest  sum  of  money  with  which  I 
can  purchase  oxen  at  $30  each,  cows  at  $60  each,  or  horses 
at  $80  each  ? 

9.  Find  the  difference  between  the  G.   C.   D.  and  the 
L.  C.  M.  of  81,  45,  108,  and  135. 

10.    What  is  the  greatest  number  that  will  exactly  divide 
3640,  12750,  and  18755  ? 


MISCELLANEOUS.  277 

431.  1.  If  the  ties  on  the  KY.C.  &  H.R.R.  are  IjJ  ft. 
apart  from  centre  to  centre,  how  many  are  there  from  New 
York  to  Buffalo,  a  distance  of  450  miles  ? 

2.  If  the  Empire  State  express  has  an  average  rate  of 
62  miles  an  hour,  how  many  hours  and  minutes  will  it 
take  to  run  from  Syracuse  to  Albany,  a  distance  of  150 
miles  ? 

3.  Multiply  7f  by  17}%. 

4.  E.  C.  Stearns  &  Co.  sell  24  bicycles  at  $62  1  apiece; 
what  do  they  bring  ? 

5.  How  many  times  does  a  bicycle  wheel  9|  ft.  in  cir- 
cumference revolve  in  going  3  miles,  there  being  5280  ft. 
in  a  mile  ? 


6.  Multiply     ^-    -f  J.  by  12  £. 

213 

7.  A  and  B  can  build  a  house  in  30  days:  B  can  do 
the  work  alone  in  45    days.     In   how  many  days  can  A 
do  it  alone  ? 

8.  Write  a  complex  fraction,  whose  numerator  shall  be 
a  simple  fraction,  and  its  denominator  compound. 

9.  A  drover  bought  375  sheep  at  $4^  per  head.     He 
sold  200  of  them  at  a  loss  of  20  cents  per  head,  and  gained 
enough  on  the  rest  to  •  balance  the   loss.     What   did   he 
receive  per  head  for  the  rest? 

10.  A  can  do  a  piece  of  work  in  5  days  ;  B  can  do  the 
same  work  in  8  days.  In  what  time  can  they  do  it  work- 
ing together  ? 

432.  1.  A  boy  paid  for  a  book  $.70,  which  was  |  of  his 
money.  The  remainder  he  spent  for  marbles  at  2^  cents 
apiece.  How  much  money  had  he  at  first  ?  and  how  many 
marbles  did  he  buy  ? 


278  SENIOR    ARITHMETIC. 

2.  At  a  school  examination  f  of  the  pupils  passed,  and 
250  pupils  failed.     How  many  pupils  were  examined  ?  and 
how  many  passed  ? 

3.  £  of  a  number  diminished  by  J  of  it  is  equal  to  5. 
What  is  the  number  ? 

4.  2\  of  1743  is  /-^  of  what  number  ? 

5.  A   man   after   giving   1,    |,  and   |   of  his  money  in 
charity  had  $10000  left.     How  much  had  he  at  first  ? 

6.  Four  persons  own  a  ship.     A  owns  \  of  it,  B  1  of 
the   remainder,   C  ^  of  what  then   remained,  and  D  the 
remainder,  which  is  worth  $3000.     What  is  the  value  of 
the  ship  ? 

7.  If  §  of  a  number  be  divided  by  4,  and  J  of  |  of  the 
number  be  taken  from  the  quotient,  the  remainder  will  be 
6.     What  is  f  of  the  number  ? 

8.  One    person   can    do    a   piece    of    work    in    6   days, 
another  can  work  twice  as  fast.     How  long  will  it  take 
them  to  do  the  work  together  ? 

9.  A   boy  was   asked   how  many  fish   he   had   caught. 
He  said  that  the  difference  between  \  and  f  the  number 
was  six.     How  many  had  he  ? 

10.  A,  B,  and  C  can  do  a  piece  of  work  in  5  da.  A  can 
do  it  alone  in  12  da.,  C  can  do  it  in  15  da. ;  in  what  time 
can  B  do  it  ? 

433.  1.  What  will  it  cost  at  $1.75  a  yard  to  carpet  a 
floor  18  ft.  long,  14  ft.  wide,  with  carpet  j  yd.  wide  ? 

2.  How  many  yards  of  carpeting  27  inches  wide  will 
be  required  for  a  room  30  ft.  long,  24  ft.  wide,  if  the  strips 
run  crosswise,  and  6  inches  be  allowed  for  matching  ? 

3.  What  fraction  of  a  great   gross  is  3  gross,   5  doz., 
1     units  ? 


279 

4.  At  $.27  per  sq.  yard,  find  the  cost  of  plastering  a 
room  30  ft.  by  24  ft.  by  12  ft.  high,  allowing  for  a  base- 
board one  foot  high,  two  doors  9  ft.  by  3  ft.,  and  5  win- 
dows 6  ft.  by  3  ft. 

5.  Reduce  5  cd.  ft.  9|  cu.  ft.  to  the  fraction  of  a  cord. 

6.  Reduce  33  gal.  3  qt.  1  pt.  1T75  gi.  to  the  fraction 
of  a  hhd. 

7.  How  much  tin  will  be  required  to  make  a  pail  and 
cover,  the  pail  to  be  6  inches  in  depth  and  7  inches  in 
diameter,  and  the  rim  of  the  cover  to  be  1  inch  deep  ? 

8.  At  $16.50  per  M.,  what  will  be  the  cost  of  12  sticks 
of  timber,  each  14  ft.  long,  10  in.  wide,  and  8  in.  thick  ? 

9.  How  many  board  feet  in  a  plank  16  ft.  long,  15  in. 
wide  at  one  end,  and  10  in.  wide  at  the  other  end,  and  3  in. 
thick  ? 

10.  The  longitude  of  New  York  is  74°  0'  3"  W.,  and  that 
of  San  Francisco  122°  23'  W.  When  it  is  1  P.M.  at  New 
York,  what  is  the  time  at  San  Francisco  ? 

434.  1.  The  longitude  of  Syracuse,  N.Y.,  is  76°  9'  16" 
W.,  and  that  of  Berlin,  Germany,  is  13°  23'  44"  E.  When 
it  is  noon  in  Berlin,  what  is  the  time  at  Syracuse  ? 

2.  The  Qswego  River  is  24  miles  long,  and  descends 
120  feet  in  that  distance.     What  is  the  average  descent 
per  mile  ? 

3.  Add  |  A.,  £  sq.  rd.,  1  sq.  yd.,  £  sq.  ft. 

4.  Find  the  cost  of  4  T.  7  cwt.  40  Ib.  of  hay  at  $12  per 
ton. 

5.  From  a  cask  containing  44  gal.  2  qt.  1  pt.  of  vine- 
gar, 8  gal.  3  qt.  leaked  out.     What  decimal  of  the  original 
contents  remained  ? 

6.  Find  the  number  of  square  inches  in  the  surface  of 
a  block  2  ft.  long,  18  in.  wide,  and  10  in.  high. 


280  SENIOR   ARITHMETIC. 

7.  The  sun  rose  in  the  latitude  of  ISfew  York,  April  1, 
1896,  at  5  o'clock  and  43  minutes,  and  set  at  6  o'clock  and 
25  minutes.     It  rose  April  30  at  4  o'clock  and  59  min- 
utes, and  set  at  6  o'clock  and   55  minutes.      How  much 
longer  was  the  thirtieth  day  than  the  first  ? 

8.  How  long  and  wide  must  a  granary  be  to  hold  4000 
bushels  of  grain,  if  it  is  8  ft.  high,  and  the  grain  to  be 
placed  in  bins  6  ft.  back  on  each  side  of  an  aisle  4  feet 
wide  ? 

9.  A  cubic   foot  of  water  weighs  62^  pounds.     How 
many  barrels   in  a  cistern  of  water  that  weighs  6  T.  5 
cwt,  ? 

10.    Find  the  cost  of  1  bu.  1  pk.  1  qt.  and  1  pt.  of  chest- 
nuts at  5/  per  quart. 

435.    1.    At  what  rate  per  cent  will  $2500  gain  $625  in 
3  years,  4  months  ? 

2.  A  merchant  buys  goods  at  $1.20  a  yard,  and,  after 
keeping  them  6  mo.,  sells  them  at  $1.35.     What  is  his  rate 
of  gain  ? 

3.  A  man  buys  oranges  at  I/  each,  and  sells  them  at 
18  cents  a  dozen.     What  is  his  gain  per  cent  ? 

4.  Find  the  amount  on  $836.22  from  Feb.  19,  1895,  to 
June  3,  189.6,  at  6%. 

5.  3200  votes  are  cast  for  two  men ;  one  has  a  majority 
of  374.     How  many  votes  did  each  receive  ? 

6.  A  man   borrowed   $756.12,   June  28,  1872.     What 
must  he  pay  to  cancel  the  debt  July  11,  1872,  at  6%  ? 

7.  A   commission    merchant    in   Minneapolis    received 
$6150,  with  directions  to  purchase  flour.     His  terms  were 
2^%    on  the   amount  purchased.      How  many  barrels   of 
flour   at   $3  a  barrel  can   he   ship  to  the  sender  of   the 
money  ? 


MISCELLANEOUS.  281 

8.  A  merchant  sells  goods  at  an  advance  of  20%,  but 
loses  5%  of  his  sales  by  bad  debts.    What  %  does  he  gain? 

9.  A  bought  a  carriage  at  20%  discount  with  10%  and 
5%  off,  and  sold  it  at  the  list  price.     What  %  profit  did 
he  inake  ? 

10.  An  agent  sold  some  Western  land,  and  paid  to  the 
former  owner  $7531.30,  retaining  $153.70  as  commission. 
What  rate  did  he  charge  ? 

436.  1.  A  district  schoolhouse  cost  $8010;  the  valua- 
tion of  the  property  of.  the  district  is  $392,375,  and  the 
number  of  polls  assessed  at  $1.25  each  is  130.  What  is 
the  rate  of  tax,  and  what  was  A's  tax,  who  paid  for  4  polls, 
the  valuation  of  his  property  being  $6000  ? 

2.  What  sum  of  money  placed  on  interest  at  6%  will 
amount  to  $1567.85  in  1  year,  3  months  ? 

3.  Sold  wheat  at  72  cents  per  bushel,  and  thereby  lost 
10  c/0  of  the  cost.     What  was  the  cost  per  bushel  ? 

4.  What  will  be  the  net  cost  of  stationery  billed  at 
$850,  if  the  discount  is  20%  and  10%  off? 

5.  A  house  worth  $7200  is  insured  for  £  of  its  value,  at 
the  rate  of  60  cents  on  $100.     Find  the  premium. 

6.  A  man  sold  a  house  for  $4200,  which  was  20%  more 
than  it  cost  him.     What  did  it  cost  ? 

7.  On  a  bill  of  goods  listed  at  $645,  choice  is  given 
between  discounts  of  20%,  10%,  and  5%   off,  or  a  direct 
discount  of  35%  off.     Which  is  better?  and  how  much  ? 

8.  If  a  merchant  gains  16^%  by  selling  cloth  at  $1.40 
per  yard,  find  his  gain  on  a  sale  amounting  to  $32. 

9.  I  owe  B  a  bill  of  $1980.     If  I  borrow  the  money 
from  a  bank,  what  must  be  the  face  of  a  note,  due  in  60  days 
without  interest,  which  I  must  give  to  the  bank,  that  I  may 
receive  the  amount  necessary  to  pay  him,  discount  at  6%  ? 


SEXIOU    ARITHMETIC. 

10.  A  man  sells  liis  house  for  $8000,  and  receives  in 
payment  a  note  for  90  days.  After  30  days  he  has  the 
note  discounted  at  a  bank  at  6%.  What  does  he  receive 
for  it  ? 

437.  1.  I  was  offered  $100  cash  for  my  buggy,  or  a  note 
of  $165  payable  in  90  days.  I  took  the  note,  and  dis- 
counted it  at  a  bank  at  5%.  Did  I  gain  or  lose  ?  and  how 
much? 

2.  What  is  the  difference  between  the  true  and  bank 
discount  on  $1250  for  90  days  at  6%  ? 

3.  If  John  lends  James  $300  for  4  months,  how  long 
ought  James  to  lend  John  $800  to  equal  the  favor  ? 

4.  I  have  a  note  of  $1225,  due  in  48  days.     Needing 
the  money  immediately,  I  get  it  discounted  at  a  bank  at 
6%.     How  much  shall  I  receive  ?  and  how  much  will  the 
bank  take  ?     No  grace. 

5.  Three  men  hire  a  pasture  for  $60.     A  put  in  4  cows 
for  11  weeks,  B  5  cows  for  12  weeks,  and  C  8  cows  for  5 
weeks.     What  ought  each  to  pay  ? 

6.  If  a  man  5  ft.  10  in.  high  casts  a  shadow  4  ft.  6  in. 
long,  what  is  the  height  of  a  tree  which  casts  a  shadow  85 
feet  long  at  the  same  time  ? 

^2  {)% 

7.  Give  the  inverse  ratio  of  -J  to  -^ . 

5  5 

8.  Required  the  ratio  of  £21  15s.  to  £6  15s. 

9.  A,  B,  and  C  entered  business  with  a  certain  capital, 
Jan.  1,  1894.     Jan.  1,  1896,  they  find  the  business  to  be 
worth  $7000,  which  is  a  gain  of  40%  on  the  original  capi- 
tal.    A's  share  of  the  gain  is  50%,  B's  share  30%,  and  C's 
share  20%.     What  amount  did  each  invest  ? 

10.    What  did  each  gain  ? 


MISCELLANEOUS.  283 

438.  1.  If  4  barrels  of  flour  will  last  3  persons  for  1 
year,  how  many  barrels  will  be  required  to  last  10  persons 
10  months  ? 

2.  The  shadow  of  a  flag-staff  at  a  certain  time  of  day 
was  64  feet  in  length.     A  line  stretched  from  the  top  of 
the  flag-staff  to  the  extremity  of  the  shadow  measured  150 
feet.     Required  the  height  of  the  staff. 

3.  Messrs.  Stevens,  Jones,  &  Payne  form  a  partnership, 
placing  into  their  business  $350,  $450,  $1500  respectively. 
They  make  $570  the  first  year.     What  share  of  the  profits 
should  each  receive  ? 

4.  By  selling  3%  stock  at  par,  and  buying  4%  at  110,  a 
man  increases  his  income  $105  a  year.     How  many  shares 
of  the  3%  stock  does  he  sell  ? 

5.  A,  B,  and  C  entered  into  a  partnership.    A  furnished 
$1200  for  8  mo.,  B  furnished  $1600  for  9  mo.,  and  C  fur- 
nished $1000  for  a  year.     They  lose  $560.     What  is  each 
man's  loss  ? 

6.  What  is  the  length  of  a  walk  laid  diagonally  through 
a  park  which  measures  60  rods  on  one  street  and  80  rods 
on  another  ? 

7.  What  will  be  the  difference  in  ratio  of  income  be- 
tween 5%  stock  bought  at  120  and  4%  bought  at  95? 

8.  A  father  dying  left  to  his  family  a  certain  sum  of 
money,  of  which  the   wife   received  $8000,  his   daughter 
$4000,  and  each  of  two  sons  $6000.     What  part  of  the 
whole  did  each  receive  ? 

9.  If  sugar  costs  5i  cents  per  pound  and  coffee  33  cents 
per  pound,  what  is  the  ratio  of  the  cost  of  the  sugar  to 
that  of  the  coffee  ? 

10.  At  $.50  per  rod,  how  much  will  it  cost  to  enclose  a 
field  of  80  acres,  that  is  twice  as  long  as  it  is  wide  ? 


284  SENIOR   ARITHMETIC. 

439.  l.    If  the  sale  of  coal  at  $.75  per  tou  above  cost 
yields  a  profit  of  18^%,  how  much  must  the  seller  add  to 
this  price  to  make  a  profit  of  40%  ? 

2.  At  what  price  must  a  4%    stock   be  purchased  to 
yield  5%   on  the  investment  ? 

3.  If  a  pile  of  wood  38  ft.  long,  4  ft.  wide,  and  5  ft. 
high  costs  $6250,  what  will  be  the  cost  of  a  pile  64  ft.  long, 
8  ft.  wide,  and  6  ft.  high  ? 

4.  If  10  men,  working  10  hr.  a  day  for  13  da.,  can  build 
a  fence  200  rd.  long,  how  many  men,  working  11  hr.  a  day 
for  10  da.,  can  build  92  rd.  of  the  same  kind  of  fence  ? 

5.  The  smaller  of  two  numbers  is  36,  and  one  half  of 
the   ratio  between  it  and  the  larger  is  2.     What  is  the 
larger  number  ? 

6.  What  number  has  the  same  ratio  to  5  that  ^  has 
to  J  ? 

7.  Find    the    mean   proportional    between   16  and  36. 
Between  T^  and  1. 

8.  What  income  on  his  investment  will  a  man  realize 
if  he  purchases  4%  stock  at  125  ? 

9.  If  A's  capital  is  $3000,  and  B's  $2000,  how  much 
more  should  B  invest  at  the  end  of  6  mo.  that  he  may 
share  equally  with  A  at  the  end  of  the  year  ? 

10.    What  is  the  rate  per  cent  of  a  tax  for  $52.88|  on 
property  assessed  at  $3525.50  ? 

440.  1.    Write  your  own  promissory  note  for  $200,  with 
interest  payable  in  ninety  days  from  to-day  to  any  person 
you  choose. 

2.  On  what  month  and  day  would  your  note  become 
due,  including  days  of  grace  ?  Give  one  reason  why  the 
note  is  void  (worthless).  Find  the  amount  due  on  your 
note  at  its  maturity. 


MISCELLANEOUS.  285 

3.  What  is  the  time  of  day  when  the  time  past  noon 
equals  the  time  to   midnight  ?     When   ^   the    time   past 
noon  equals  the  time  to  midnight  ?     When  the  time  past 
noon  equals  i  the  time  to  midnight  ? 

4.  A  cask  can  be  emptied  by  a  ^-inch  faucet  in  4  hours. 
In  what  time  can  it  be  emptied  by  a  1^-inch  faucet  ? 

5.  Explain  the  difference  between  factor  and  root ;  be- 
tween product  and  power. 

6.  A  and  B  divide  $90  in  the  ratio  of  §  to  |.     What  is 
each  one's  share  ? 

7.  If  a  tank  13^  ft.  long,  1\  ft.  wide,  and  3^  ft.  deep 
holds  73^  barrels  of  water,  how  wide  must  another  tank  be 
that  is  9  ft.  9  in.  long,  4  ft.  10  in.  deep,  and  holds  89£ 
barrels  ? 

8   1  +  ft)'  -  "fr*  -  •> 

i  x  ay 

9.  A  milkman's  quart  measure  is  too  small  by  one  gill. 
At  5  cents  a  quart,  how  much  does  he  dishonestly  make  in 
the  month  of  June,  if  he  sells  500  false  quarts  daily  ? 

10.    Find  the  length  of  the  diagonal  of  an  are  of  land  in 
the  form  of  a  square. 

MENSURATION. 

441.  The  process  of  measuring  lines,  surfaces,  and  solids 
is  Mensuration. 

442.  A  Line  is  that  which  has  length,  without  breadth 
and  thickness. 

443.  A  Straight  Line  is  the  shortest  distance  between  two 
points,  or  a  line  that  does  not  change  its  direction  at  any 
point. 

444.  A  Curved  Line  changes  its  direction  at  every  point. 


286  SENIOR   AK1THMETIC. 

445.  A  Plane  Surface  is  a  surface  that  does  not  change 
its  direction. 

446.  A  Quadrilateral  is  a  plain  figure  having  four  straight 
sides. 

447.  Parallel  Lines  are  lines  having  the  same  direction 
and  equally  distant  from  each  other. 

448.  A  Parallelogram  is  a  quadrilateral  whose  opposite 
sides  are  parallel. 

449.  A  Rhomboid  is  a  parallelogram  whose  angles  are 
not  right  angles. 

What  is  a  parallelogram  called  whose  angles  are  right 
angles  ? 

450.  A  Rhombus  is  a  rhomboid  whose  sides  are  equal. 


A  Rhomboid.  A  Rhombus. 

451.    The  area  of  a  rhomboid  is  found  by  multiplying  the 
base  by  the  altitude. 

NOTE.  —  The  altitude  of  a  parallelogram  is  the  perpendicular 
distance  between  the  sides. 

1.  Find  the  area  of  a  parallelogram  whose  base  is  24 
rods,  and  altitude  18  rods. 

2.  Find  the  area  of  a  rhombus  whose  base  is  15  ft.  and 
altitude  8  ft. 

3.  Draw  a  rhomboid  whose  base  is  15  ft.  and  altitude 
10  ft.     Find  its  area. 

Draw  a  rectangle  having  the  same  dimensions. 


MENSURATION. 


287 


A  Trapezoid. 


452.  A  Trapezoid  is   a   quadrilateral   having   only  two 

sides  parallel. 

453.  To  find  the  area  of  a  trap- 
ezoid,  multiply  £  the  sum  of  the 
parallel  sides  by  the  altitude. 

4.  Find  the  area  of  a  trap- 
ezoid  whose  altitude  is  10  ft.,  its 

longest  side  20  ft.,  and  shortest  side  15  f t.  ? 

5.  A  board  20  inches  wide  at  one  end  and  12  inches 
wide  at  the  other  is  16  feet  long.     How  many  board  feet 
does  it  contain  ? 

454.  A  Trapezium  is  a  quad- 
rilateral having  no  two  sides 
parallel. 


NOTE.  —  By  drawing  a  diagonal 
between  any  two  opposite  sides  of 
a  trapezium,  we  have  two  triangles, 
the  diagonal  serving  as  the  hase  of 
each.  The  altitude  of  each  is  the 
perpendicular  distance  from  its  other  angle  to  the  diagonal. 


A  Trapezium. 


455.  To  find  the  area  of  a  trapezium,  multiply  the  diago- 
nal by  half  the  sum  of  the  altitudes  of  the  two  triangles. 

6.  The  diagonal  of  a  trapezium  is  18  ft. ;  the  altitudes 
of  its  two  triangles  are  5  ft.  and  3  ft.     What  is  the  area  ? 

7.  A  farm  is  in  the  form  of  a  trapezium.    The  diagonal 
distance  between  the  northern  and  southern  corners  is  108 
rods,  and  the  perpendicular  distances  from  the  east  and 
west  corners  to  the  diagonal  are  52    rods    and   36    rods 
respectively.     How  many  acres  in  the  farm  ? 

SOLIDS. 

456.  A  solid  whose  two  bases  are  equal  and  parallel, 
and  its  other  faces  parallelograms,  is  called  a  Prism. 


288 


SENIOR    ARITHMETIC. 


A  Triangular 
Prism. 


A  Rectangular 
Prism. 


NOTE.  —  Prisms  take  their  names  from  the  form  of  their  bases,  as 
triangular,  quadrangular,  pentagonal,  hexagonal,  etc.,  according  as 
the  bases  have  three,  four,  five,  or  six  sides,  etc. 

457.  To  find  the  contents  of  a 
prism,  multiply  the  area  of  the 
base  by  the  altitude. 

8.  Find  the  contents  of  a  tri- 
angular prism  .whose  altitude  is 
10  in.,  and  the  area  of  its  base 
7  sq.  in. 

9.  What  are  the  contents  of 
a   quadrangular    prism    whose 

base  is  5  in.  by  8  in.,  and  whose  altitude  is  12  in.  ? 

10.  What  are  the  contents  of  a  hexagonal  prism,  the  area 
of  whose  base  is  10  sq.  ft.,  and  whose  altitude  is  15  ft.  ? 

PYRAMIDS   AND   CONES. 

458.  A  solid  whose  base  is  a  triangle,  square,  pentagon, 
etc.,  and  whose  sides  are  triangles  meeting  at  a  vertex,  is 
called  a  Pyramid. 

NOTE.  —  A  pyramid  takes  its  name  from  the  form  of  its  base. 

A  solid  whose  base  is  a  circle,  and  whose  convex  surface 
terminates  in  a  point,  is  called  a  Cone. 

459.  The  Altitude  of  a  pyr- 
amid or  cone  is  the  perpendic- 
ular distance  from  its  vertex 
to  the  centre  of  its  base. 

The  Slant  Height  is  the 
shortest  distance  from  the 
vertex  to  the  perimeter  of 
the  base. 

460.  To  find  the  contents  of  a  pyramid  or  cone,  multiply 
the  area  of  the  base  by  $  of  the  altitude. 


A  Pyramid. 


A  Cone. 


MENSURATION.  289 

To  find  the  convex  surface,  multiply  the   perimeter  of 
the  base  by  ^  the  slant  height. 

11.  Find  the  contents  of  a  quadrangular  pyramid  whose 
altitude  is  10  in.,  and  whose  sides  of  bases  are  8  in.  and 
6  in. 

SOLUTION.  —  8  x  6  x  V1  =  16°  cu  in-     ^ns. 

12.  Find   the   convex    surface    of    a   regular  hexagonal 
pyramid  whose  slant  height  is  16  in.,  and  whose  side  of 
base  is  4  in. 

13.  Find  the  convex  surface  of  a  cone,  when  the  circum- 
ference of  its  base  equals  16  ft.  and  its  slant  height  18  ft. 

14.  Find  the  convex  surface  and  volume  of  a  cone  whose 
radius  is  4  in.  and  altitude  6  in. 

461.  The  Frustum  of  a  cone   or  pyramid   is   the   part 
which  is  left  after  the  top  is  cut  off  in  a  plane  parallel 
to  the  base. 

462.  To  find  the  contents  of  the  frustum  of  a  pyramid  or 
cone,  multiply  i  of 

the  altitude  by  the 
sum  of  the  areas 
of  the  two  bases 
plus  the  square 
root  of  their  pro- 
duct. 

15.  Find  the  COn-          Frustum  of  a  Pyramid.  Frustum  of  a  Cone. 

tents  of   the  frus- 
tum of  a  quadrangular  pyramid  whose  altitude  is  15  ft., 
and  whose  ends  are  6  ft.  and  4  ft.  square. 

16.  A  log  16  ft.  long  is  30  in.  in  diameter  at  one  end 
and  24  in.  at  the  other.     Find  its  cubical  contents. 

463.  A  Sphere  is  a  solid  bounded  by  a  curved  surface, 
all  parts  of  which  are  equally  distant  from  the  centre. 


290  SENIOR   ARITHMETIC. 

464.    To  find  the  surface  of  a  sphere,  multiply  the  circum- 
ference by  the  diameter. 

465.  To  find  the  contents  of  a  sphere, 
multiply  the  surface  by  £  of  the 
diameter. 

17.  Find  the  surface  of   a  sphere 
when  the  diameter  is  16  inches. 

18.  Find  the  surface  of  a  sphere 
when  the  radius  is  3  yards. 

19.  Find  the  surface  of  a  sphere 
when  the  radius  is  5  cm. 

Find  the  volume  when : 

20.  Diameter  =  25  ft.      22.    Kadius  =  12  ft. 

21.  Kadius  =  2  ft.  23.    Circumference  =  125664  in. 

24.  Radius  =  3  dm. 

25.  Compare  the  volume   of   a  4-ft.    cube   and   a   4-ft. 
sphere. 

26.  Compare  the  surfaces  of   a  4-ft.   cube  and  a  4-ft. 
sphere. 

REVIEW  OF  MENSURATION. 

466.    1.    Find  the  cubic  yards  in  a  cone,  the  circumfer- 
ence of  whose  base  is  20  ft.,  and  whose  altitude  is  30  ft. 

2.  Find  area  of  a  semi-circle  when  its  radius   equals 
14  ft. 

3.  Find  area  of  a  square  inscribed  in  a  circle  of  4  ft. 
in  diameter. 

4.  The  circumference  of  a  circle  and  the  perimeter  of  a 
square  are  each  300  ft.     Which  has  the  greater  area  ? 

5.  A  circle  is  inscribed  in  a  6-ft.  square.     Find  the  area 
of  the  circle. 


MENSURATION.  291 

6.  Find  the  value  at  $50  an  acre  of  a  farm  in  the  form 
of  a  trapezoid,  the  parallel  sides  of  which  are  120  rd.  and 
160  rd.  respectively,  the  distance  between  which  is  80  rd. 

7.  How  many  miles  does  the  earth  travel  in  a  revolu- 
tion  around   the   sun,  the  distance  between   them   being 
95,000,000  miles? 

8.  If  a  bin  is  8  feet  square,  how  deep  must  it  be  to 
hold  100  bushels  ? 

9.  Find  the  lateral  surface  of  an  equilateral  triangular 
pyramid,  the  perimeter  of  the  base  being  12  m.  and  the 
slant  height  14  m. 

10.  Find  the  volume  of  a  square  pyramid,  the  perimeter 
of  whose  base  is  16  feet,  and  whose  altitude  is  9  ft. 

11.  What  is  the  volume  of  the  largest  cone  that  can  be 
cut  from  a  pyramid  whose  base  is  6  feet  square,  and  whose 
slant  height  is  15  feet  ? 

12.  A  cylindrical  tank  is  14  ft.  deep  and  6  feet  in  diam- 
eter.    Find  the  cost  of  cementing  sides  at  90/  a  sq.  yard. 

13.  Find  the  capacity  in  gallons  of  a  cylindrical  cistern 
whose  inside  diameter  is  6  feet,  and  whose  depth  is  7  feet. 

14.  Find  the  capacity  in  Kl.   of   a  cylindrical  cistern 
whose   inside   diameter   is   4   m.,   and  whose   altitude   is 
5  m. 


YB   17452 


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